Abstract: Publication date: Available online 12 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Thorben Casper, Ulrich Römer, Herbert De Gersem, Sebastian Schöps This work addresses the simulation of heat flow and electric currents in thin wires. An important application is the use of bond wires in microelectronic chip packaging. The heat distribution is modeled by an electrothermal coupled problem, which poses numerical challenges due to the presence of different geometric scales. The necessity of very fine grids is relaxed by solving and embedding a 1D sub-problem along the wire into the surrounding 3D geometry. The arising singularities are described using de Rham currents. It is shown that the problem is related to fluid flow in porous 3D media with 1D fractures (C. D’Angelo, SIAM Journal of Numerical Analysis 50.1, pp. 194–215, 2012). A careful formulation of the 1D–3D coupling condition is essential to obtain a stable scheme that yields a physical solution. Elliptic model problems are used to investigate the numerical errors and the corresponding convergence rates. Additionally, the transient electrothermal simulation of a simplified microelectronic chip package as used in industrial applications is presented.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Pengfei Xu, Guang-an Zou, Jianhua Huang The current paper is devoted to the time–space fractional Ginzburg–Landau equation driven by fractional Brownian motion. The spatial–temporal regularity of the nonlocal stochastic convolution is firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag-Leffler functions. Finally, the numerical simulations for the time-fractional stochastic Ginzburg–Landau equation are provided to verify the mathematical analysis results.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Huimin Wang This work reports a study of Klein–Gordon–Zakharov equation by using the lattice Boltzmann method. The Klein–Gordon–Zakharov equation describes the interaction of the Langmuir wave and the ion acoustic wave in plasma. Numerical simulations of the constructed lattice Boltzmann model for the Klein–Gordon–Zakharov equation show the propagation and the interaction of the (1+1) and (2+1) dimensional solitary waves. These simulation results are examined and compared by using different parameters and different schemes. This work indicates that the lattice Boltzmann method can be a very effective tool for the simulation of Klein–Gordon–Zakharov equation.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Thirupathi Gudi, Papri Majumder In this article, we propose and analyze conforming and discontinuous Galerkin (DG) finite element methods for numerical approximation of the solution of the parabolic variational inequality associated with a general obstacle in Rd(d=2,3). For fully discrete conforming method, we use globally continuous and piecewise linear finite element space. Whereas for the fully-discrete DG scheme, we employ piecewise linear finite element space for spatial discretization. The time discretization has been done by using the implicit backward Euler method. We present the error analysis for the conforming and the DG fully discrete schemes and derive an error estimate of optimal order O(h+Δt) in a certain energy norm defined precisely in the article. The analysis is performed without any assumptions on the speed of propagation of the free boundary but only assumes the pragmatic regularity that ut∈L2(0,T;L2(Ω)). The obstacle constraints are incorporated at the Lagrange nodes of the triangular mesh and the analysis exploits the Lagrange interpolation. We present some numerical experiment to illustrate the performance of the proposed methods.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Saumya Bajpai, Ambit K. Pany In this article, a finite element Galerkin method is applied to the Kelvin–Voigt viscoelastic fluid model, when its forcing function is in L∞(L2). Some new a priori bounds for the velocity as well as for the pressure are derived which are independent of inverse powers of the retardation time κ. The second order error estimate in L∞(L2)-norm, the first order error estimate in L∞(H01)-norm for the velocity and the first order error estimate in L∞(L2)-norm for the pressure for the semidiscrete method are derived which hold uniformly with respect to κ as κ→0 with the initial condition only in H2∩H01. Further under the smallness assumption on the data, these error estimates are shown to be uniform in time as t↦∞. For the complete discretization of the semidiscrete system, a first-order accurate backward Euler method is applied and fully discrete error estimates are established. Finally, numerical experiments are conducted to verify the theoretical results. The results derived in this article are sharper than those derived earlier for finite element analysis of the Kelvin–Voigt fluid model in the sense that the error estimates in this article hold true uniformly even as κ→0.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Li-Feng Yin, Xing-Ping Wu In this paper, we study the following Schrödinger equation −Δu+(1+μa(x))u=f(u)+ u 2∗−2u,inRN,where N≥3 and μ>0. We obtain a ground state solution for the above equation. Moreover, the concentration behavior of the ground state solution is also described as μ→∞.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): S. Falletta, G. Monegato, L. Scuderi We consider (transient) 3D elastic wave propagation problems in unbounded isotropic homogeneous media, which can be reduced to corresponding 2D ones. For their solution, we propose and compare two boundary integral equation approaches, both based on the coupling of a discrete time convolution quadrature with a classical space collocation discretization. In the first approach, the PDE problem is preliminarily replaced by the equivalent well known (vector) space–time boundary integral equation formulation, while in the second, the same PDE is replaced by a system of two (coupled) wave equations, each one of which is then represented by the associated boundary integral equation. The construction of these two approaches is described and discussed. Some numerical testing are also presented.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Vikram V. Garg, Serge Prudhomme We analyze the impact of using the penalty method on the estimation of a prescribed solution functional or ‘Quantity of Interest’ (QoI). Specifically, we consider the use of penalty methods to enforce Dirichlet boundary constraints, focusing our attention on boundary fluxes as QoIs. We propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter ϵ. We show that the new estimator reduces the error arising from the use of the penalty method from O(ϵ) to o(ϵ). A well-posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method. Errors in the enhanced flux estimator are then controlled using adjoint-based techniques. Several numerical experiments are presented to demonstrate that the enhanced estimator, in combination with adjoint error estimation, allows one to efficiently control both the discretization and penalty errors in target QoIs.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Leijie Qiao, Da Xu In this paper, a second-order backward differentiation formula (BDF) alternatingdirection implicit (ADI) orthogonal spline collocation (OSC) method is presented for the fractional integro-differential equation with two different weakly singular kernels. The integral terms are approximated by the second-order fractional quadrature rule suggested by Lubich. The convergence of the BDF ADI OSC method in L2 norm is proved. Besides, we provide the numerical experiments to demonstrate the results of theoretical analysis and show the accuracy and the effectiveness of the BDF ADI OSC method. Numerical experiments also exhibit the optimal error estimates.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Fabio Sanchez, Juan G. Calvo, Esteban Segura, Zhilan Feng We formulate an age-structured three-staged nonlinear partial differential equation model that features nonlinear recidivism to the infected (infectious) class from the temporarily recovered class. Equilibria are computed, as well as local and global stability of the infection-free equilibrium. As a result, a backward-bifurcation exists under necessary and sufficient conditions. A generalized numerical framework is established and numerical experiments are explored for two positive solutions to exist in the infectious class.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Nawdha Thakoor The radial point interpolation method is increasingly being applied for the numerical solution of partial differential equations in different fields. Most implementations in the literature for obtaining the matrix coefficients make use of numerical approximations of the shape functions and their derivatives. To avoid the solution of linear systems required for computation of derivative approximations, this work derives analytical shape functions for three and five-node support domains in a local radial point interpolation method (LRPIM) with multiquadrics as basis functions. A weak form algorithm for the Black–Scholes equation using three-node analytical shape functions is developed and its unconditional stability and convergence are theoretically established. LRPIM finite-difference (FD) formulas are derived and applied to the solution of one and two-asset financial options. A five-node LRPIM-FD method in one-dimension is shown to yield fourth-order accuracy and applications to two-asset problems also yield accurate prices for options on a minimum of two risky assets and exchange options.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Kaveh Fardipour, Kamyar Mansour In this paper we present a modified seventh-order weighted essentially nonoscillatory scheme for hyperbolic conservation laws. Local smoothness indicators are constructed based upon Lagrange’s interpolation polynomial. We constructed a new high-order global smoothness indicator to guarantee the scheme achieves optimal order of accuracy at critical points. We investigated this scheme at critical points and verified its order of convergence with the help of linear scalar test cases. We implemented it to various nonlinear scalar equations and system of Euler equations in one- and two-dimensions to demonstrate the discontinuity capturing and high resolution properties of the modified scheme.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): Patricio Farrell, Dirk Peschka We study different discretizations of the van Roosbroeck system for charge transport in bulk semiconductor devices that can handle nonlinear diffusion. Three common challenges corrupting the precision of numerical solutions will be discussed: boundary layers, discontinuities in the doping profile, and corner singularities in L-shaped domains. We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as advanced Scharfetter–Gummel type finite-volume discretization schemes. The most problematic of these challenges are boundary layers in the quasi-Fermi potentials near ohmic contacts, which can have a drastic impact on the convergence order. Using a novel formal asymptotic expansion, our theoretical analysis reveals that these boundary layers are logarithmic and significantly shorter than the Debye length.

Abstract: Publication date: 15 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 12Author(s): M. Hosseininia, M.H. Heydari, F.M. Maalek Ghaini, Z. Avazzadeh In this study, we introduce the nonlinear variable-order time fractional two-dimensional (2D) Klein–Gordon equation by using the concept of variable-order fractional derivatives. The variable-order fractional derivative operator is defined in the Caputo type. Due to the useful properties of wavelets, we propose an efficient semi-discrete method based on the 2D Legendre wavelets (LWs) to numerically solve this equation. In fact, according to the proposed method, the variable-order time fractional derivative should be discretized in the first stage, and then the solution of the problem expanded in terms of the 2D LWs. However, the main objective of this study is to illustrate that the 2D LWs can be a useful tool for solving the nonlinear variable-order time fractional 2D Klein–Gordon equation. Stability analysis of the presented method is investigated theoretically and numerically. Moreover, the applicability and accuracy of the method are investigated by solving some numerical examples. Numerical results confirm the spectral accuracy of the established method.

Abstract: Publication date: Available online 9 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Maria Chiara D’Autilia, Ivonne Sgura, Valeria Simoncini Systems of reaction–diffusion partial differential equations (RD-PDEs) are widely applied for modeling life science and physico-chemical phenomena. In particular, the coupling between diffusion and nonlinear kinetics can lead to the so-called Turing instability, giving rise to a variety of spatial patterns (like labyrinths, spots, stripes, etc.) attained as steady state solutions for large time intervals. To capture the morphological peculiarities of the pattern itself, a very fine space discretization may be required, limiting the use of standard (vector-based) ODE solvers in time because of excessive computational costs. By exploiting the structure of the diffusion matrix, we show that matrix-based versions of time integrators, such as Implicit–Explicit (IMEX) and exponential schemes, allow for much finer problem discretizations. We illustrate our findings by numerically solving the Schnakenberg model, prototype of RD-PDE systems with Turing pattern solutions, and the DIB-morphochemical model describing metal growth during battery charging processes.

Abstract: Publication date: Available online 7 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Yong Wang, Chengwen Zhong, Jun Cao, Congshan Zhuo, Sha Liu In this paper, the original finite volume lattice Boltzmann method (FVLBM) on an unstructured grid (Part I of these twin papers) is extended to simulate turbulent flows. To model the turbulent effect, the k−ω SST turbulence model is incorporated into the present FVLBM framework and is also solved by the finite volume method. Based on the eddy viscosity hypothesis, the eddy viscosity is computed from the solution of the k−ω SST model, and the total viscosity is modified by adding this eddy viscosity to the laminar (kinematic) viscosity given in the Bhatnagar–Gross–Krook collision term. Because of solving for the collision term with the explicit method in the original FVLBM scheme, the computational efficiency is much lower for simulating high Reynolds number flow. This is due to the fact that the largest time step decided by the stability condition of the collision term, which is less than twice the relaxation time, is much smaller than that decided by the CFL condition. In order to enhance the computational efficiency, the three-stage second-order implicit–explicit (IMEX) Runge–Kutta method is used for temporal discretization, and the time step can be one or two orders of magnitude larger as compared with the explicit Euler forward scheme. Although the computational cost is increased, the final computational efficiency is enhanced by about one-order of magnitude and good results can also be obtained at a large time step through the test case of a lid-driven cavity flow. Two turbulent flow cases are carried out to validate the present method, including flow over a backward-facing step and flow around a NACA0012 airfoil. The numerical results are found to be in agreement with experimental data and numerical solutions, demonstrating the applicability of the present FVLBM coupled with the k−ω SST model to accurately predict the incompressible turbulent flows.

Abstract: Publication date: Available online 5 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): E. Artioli, A. Sommariva, M. Vianello We compute low-cardinality algebraic cubature formulas on convex or concave polygonal elements with a circular edge, by subdivision into circular quadrangles, blending formulas via subperiodic trigonometric Gaussian quadrature and final compression via Caratheodory–Tchakaloff subsampling of discrete measures. We also discuss applications to the VEM (Virtual Element Method) in computational mechanics problems.

2
O

Abstract: Publication date: Available online 2 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Ramin Mashayekhi, Hossein Arasteh, Davood Toghraie, S. Hossein Motaharpour, Amir Keshmiri, Masoud Afrand This is a numerical study of convective heat transfer of the Water-Al2O3 nanofluid in an oval channel using two-phase mixture model. The channel is fitted with two rows of twisted conical strip inserts with various directions relative to each other leading to three different combinations of the mentioned inserts, namely inward Co-Conical inserts (CCI-inward), Counter-Conical inserts (CoCI), and outward Co-Conical inserts (CCI-outward) in which its lower wall is exposed to a constant heat flux. The effect of Reynolds number ranging from 250 to 1000, nanofluid volume fraction ranging from 1 to 3 % and conical strip insert combinations are examined on the fluid flow patterns and heat transfer characteristics. The results showed that among the three combinations of the twisted conical strip insert, CCI-inward locally presents the highest values of heat transfer coefficient, as about 17% higher than plain tube, considering the nature of the secondary flow created in this case. It is also found that the effect of increasing nanofluid concentration on the channel thermal performance is more significant at higher values of Re number; however, the pressure drop difference between the three models is subtle.

Abstract: Publication date: Available online 2 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Rasool Alizadeh, Sina Rezaei Gomari, Ahmad Alizadeh, Nader Karimi, Larry K.B. Li The transport of heat and mass from the surface of a cylinder coated with a catalyst and subject to an impinging flow of a Casson rheological fluid is investigated. The cylinder features circumferentially non-uniform transpiration and is embedded inside a homogeneous porous medium. The non-equilibrium thermodynamics of the problem, including Soret and Dufour effects and local thermal non-equilibrium in the porous medium, are considered. Through the introduction of similarity variables, the governing equations are reduced to a set of non-linear ordinary differential equations which are subsequently solved numerically. This results in the prediction of hydrodynamic, temperature, concentration and entropy generation fields, as well as local and average Nusselt, Sherwood and Bejan numbers. It is shown that, for low values of the Casson parameter and thus strong non-Newtonian behaviour, the porous system has a significant tendency towards maintaining local thermal equilibrium. Furthermore, the results show a major reduction in the average Nusselt number during the transition from Newtonian to non-Newtonian fluid, while the reduction in the Sherwood number is less pronounced. It is also demonstrated that flow, thermal and mass transfer irreversibilities are significantly affected by the fluid’s strengthened non-Newtonian characteristics. The physical reasons for these behaviours are discussed by exploring the influence of the Casson parameter and other pertinent factors upon the thickness of thermal and concentration boundary layers. It is noted that this study is the first systematic investigation of the stagnation-point flow of Casson fluid in cylindrical porous media.

Abstract: Publication date: Available online 2 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Mohammadreza DaqiqShirazi, Azeez A. Barzinjy, Samir M. Hamad, Rezvan Alamian, Mostafa Safdari Shadloo In the present work, a transient numerical study on two-phase flow over circular staggered and inline micro pin fins is conducted. Pin diameter, fluid density ratio, and surface tension have been taken into account. Diameters of micro pins are in the range of 50 to 250μm; moreover, a variation of 100 fold in the surface tension and density ratio are considered. In order to solve the numerical model of a micro heat sink, a volume of fluid (VOF) method using open package Gerris Flow Solver (GFS) is employed. Pressure drop, flow velocity, as well as void fraction for all cases are presented and discussed. Based on the results, transient nature of flow is observed even after the first transition phase. Flow mixing as an essential phenomenon in heat transfer is thoroughly discussed. Our study proves that with an increase of pin diameter the flow mixing near micropins is hindered. Moreover, bridge formation in staggered conformation was observed which may reduce the heat transfer. Additionally higher surfaces tension ratio yielded a better flow mixing.

Abstract: Publication date: Available online 1 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Seyed Mahmood Mousavi, Reza Kamali, Freshteh Sotoudeh, Nader Karimi, Danial Khojasteh In this paper, the Pseudo shock structure in a convergent–long divergent duct is investigated using large eddy simulation on the basis of Smagorinsky–Lilly, Wall-Adapting Local Eddy-Viscosity and Algebraic Wall-Modeled LES subgrid models. The first objective of the study is to apply different subgrid models to predict the structure of Lambda form shocks system, while the ultimate aim is to obtain further control of the shock behavior. To achieve these goals, the dynamic grid adaption and hybrid initialization techniques are applied under the 3D investigation to reduce numerical errors and computational costs. The results are compared to the existing experimental data and it is found that the WMLES subgrid model results in more accurate predictions when compared to the other subgrid models. Subsequently, the influences of the divergent section length with the constant ratio of the outlet to throat area and, the effects of discontinuity of the wall temperature on the flow physics are investigated. The results indicate that the structure of compressible flow in the duct is affected by varying these parameters. This is then further discussed to provide a deeper physical understanding of the mechanism of Pseudo shock motion.

p
-+and+

Abstract: Publication date: Available online 31 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): O.Čertík, F. Gardini, G. Manzini, L. Mascotto, G. Vacca We discuss the p- and hp-versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schrödinger equation with a pseudo-potential term. As an interesting byproduct, we present for the first time in literature an explicit construction of the stabilization of the mass matrix. We present in detail the analysis of the p-version of the method, proving exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing hp-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the hp-spaces.

Abstract: Publication date: Available online 31 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Vivekanand Shukla, Jeeoot Singh Angle-ply laminated plates have been examined for flexure analysis with multiquadric radial basis function (MQRBF) base meshfree method. A new higher order shear deformation theory (HSDT) representing the transverse shear strain function is proposed in the displacement field. Proposed theory satisfies the condition of continuity and differentiability and is accurate enough to predict the results of flexure. Governing differential equations (GDEs) are derived using energy principle. A MATLAB code is developed to obtain the results for deflection and stresses of the angle-ply laminated plate. The flexure results for laminated plates under patch loads commonly used in general practice is obtained with proposed HSDTs and meshfree method. Present results are validated with published results for the cross-ply laminated plate, and some new results are produced for different loading conditions for angle-ply laminated plates. In the present study, eight layered symmetric cross-ply laminate of equal thickness, an antisymmetric cross-ply laminate of equal thickness and a general angle-ply laminate of equal thickness is considered. The effect of the span to thickness ratio, number of layers and shifting patch loads is presented.

Abstract: Publication date: Available online 29 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Yiran Xu, Jingye Li, Xiaohong Chen, Guofei Pang We propose a radial basis function collocation method (RBF method) to solve fractional Laplacian visco-acoustic wave equation for the Earth media having heterogeneous velocity model and complex geometry. Unlike the fractional Laplacian wave equation proposed in Zhu and Harris (2014), the wave equation we consider has a different definition for the fractional Laplacian. Specifically, spectral and Riesz fractional Laplacians are considered in Zhu and Harris (2014) and the present paper, respectively. Accordingly, the Fourier pseudospectral method (FPS method) and the RBF method are employed to solve the spectral and the Riesz fractional Laplacian wave equations. The two wave equations are observed to produce obviously different wavefields. We demonstrate the validity and flexibility of the proposed RBF method by considering five benchmarks of seismic forward modeling: (1) two-dimensional Earth media with four types of velocity models (homogeneous, two-layer, homogeneous but complex-geometry, and heterogeneous models) and (2) a spherical medium with homogeneous velocity model. We make a three-way comparison among numerical solutions to the Riesz fractional Laplacian, the spectral fractional Laplacian, and the integer-order visco-acoustic wave equations, and observe that when wave attenuation is weak the Riesz wave equation yields more similar wavefield to that of the integer-order wave equation than the spectral wave equation does. Furthermore, uniform and quasi-uniform layouts for collocation points of the RBF method are considered, and the latter layout turns out to be economical since it can preserve the solution accuracy with the minimum number of collocation points. The RBF method is truly mesh-free and dimension-free and can easily handle high-dimensional, irregular domains. Additionally, the method is easier to implement than element-based methods, such as finite element and spectral element methods, for discretizing the Riesz fractional Laplacian.

Abstract: Publication date: Available online 29 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Wenke Zhao, Xin Chen, Yaning Zhang, Wentao Su, Fei Xu, Bingxi Li The deicing performances of a road unit driven by a hydronic heating system in severely cold regions of China were investigated by using the Open Source Lattice Boltzmann Code. The model used the enthalpy-based method and double distribution functions for the velocity and temperature fields to solve the ice melting problem. The average road surface temperatures and melting conditions (melting ratio and mass) as affected by ice thickness (3 ∼20 mm), pipe spacing (80 ∼240 mm), and air temperature (252 ∼268 K) were detailed, and the parameters were analyzed by the orthogonal test method. The results show that for the ice thickness increasing from 3 mm to 20 mm, the heating rate was slightly increased from 6.3 K/h to 6.45 K/h in the preheating and initial melting stages, and from 1.32 K/h to 2.06 K/h in the rapid melting stage, the melting mass was increased from 360 g/m to 762.06 g/m whereas the melting ratio was decreased from 1.00 to 0.32. The pipe spacing of 120 mm was suitable for the road heating whereas ≥ 160 mm was not feasible in the severely cold regions. For the air temperature was increased from 252 K to 268 K, the preheating time was decreased from 3.02 h to 0.82 h, the heating rate in the melting stages was increased from 0.68 K/h to 2.94 K/h and the melting ratio was linearly increased. The parameter analysis reveals that the air temperature had the most important influence on the melting ratio whereas the pipe spacing had the most significant impact on the average road surface temperature, and the ice thickness had a slight influence on the melting ratio and average road surface temperature.

Abstract: Publication date: Available online 29 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Yao Yu, Guangwei Yuan, Zhiqiang Sheng, Yonghai Li We construct a new scheme whose primary unknowns include both cell-centered unknowns and edge unknowns on discontinuous line to solve diffusion equations with discontinuous coefficient. First, two linear fluxes are given on both sides of the cell edge, respectively. In order to deal with the defect of the existing scheme preserving maximum principle for solving diffusion problem with discontinuous coefficient, in addition to cell-centered unknowns, we also introduce cell-edge unknowns on discontinuous line as basic unknowns. Second, the conservative flux is constructed by using nonlinear weighted combination of these two linear fluxes. For the cell-edge unknowns on discontinuous line, we add an equation by using the continuity of normal flux. Compared to the classical cell-centered nonlinear finite volume scheme, the introduction of cell-edge unknowns on discontinuous edge is the key point for our scheme to solve diffusion equations with discontinuous coefficient. Then we prove that the scheme satisfies the discrete maximum principle. Based on this, the existence of a solution for the scheme is also obtained. Numerical results are presented to show that our scheme obtains almost second order accuracy for solution on random meshes, preserves discrete maximum principle, and is superior to the existing scheme (in Sheng and Yuan (2011)) preserving the maximum principle on dealing with the problems with strong anisotropic discontinuous coefficient on distorted meshes.

Abstract: Publication date: Available online 26 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Yousef Kazemian, Saman Rashidi, Javad Abolfazli Esfahani, Omid Samimi-Abianeh The Lattice Boltzmann method is used to simulate the propane–air mixture combustion in a porous media with different grain shapes of triangular, elliptical, rectangular, and star. The effects of these shapes on the flow and temperature fields and the flame characteristics were investigated. In order to simulate the flow in the porous medium, the method of creating barriers against the flow is used. The black and white photos are transformed into the matrix of 0 and 1, written in the form of the lattice Boltzmann code for solving the momentum, energy, and concentration equations. The results show that the porous media made by grains with sharp corners provide the highest reverse flow. The porous media with grain shapes of rectangular and triangular provide the highest and least pressure drop, respectively. The length of the flame decreases by using the porous media and the star shape has the largest length of the flame, while the rectangular one has the smallest flame length. The shapes of flames obtained for the elliptical and triangular grains are very similar to each other, while, the shape of flame for the star grain is close to the non-porous medium case. In general, using the porous media increases the heat transfer rate from the walls and also creates the fluctuations in the heat transfer along the wall.

Abstract: Publication date: Available online 25 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Youngho Min, Seungil Kim In this paper, we study a multiple Dirichlet-to-Neumann (MDtN) boundary condition for solving a time-harmonic multiple scattering problem governed by the Helmholtz equation in waveguides that include multiple obstacles, cavities or inhomogeneities with straight waveguides placed between them. The MDtN condition is derived by analyzing analytic solutions represented by Fourier series in the straight waveguides between obstacles, cavities or inhomogeneities. The proposed method is then to remove the straight waveguides between scatterers and impose the MDtN condition on artificial boundaries resulting from domain truncation. This numerical technique can allow a great reduction of computational efforts. The well-posedness of the reduced problem with the full MDtN condition and the reduced problem with truncated MDtN conditions are established. Also the exponential convergence of approximate solutions satisfying truncated MDtN conditions will be proved.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Renhai Wang, Bixiang Wang This article is concerned with the asymptotic behavior of solutions of a wide class of non-autonomous, non-local, fractional, stochastic p-Laplacian equations driven by multiplicative white noise. The time-dependent nonlinear drift term of the equation has a polynomial growth of arbitrary order in its third component which is allowed to be greater than the exponent p. We first employ the Faedo–Galerkin method to prove the well-posedness of the equation in an appropriate Hilbert space. We then establish the existence, uniqueness and periodicity of tempered pullback random attractors for the equations. The upper semi-continuity of these attractors is also derived as the density of noise tends to zero. The results of this paper are new even when the stochastic equation reduces to the deterministic one.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Rui Zhang, Xiangchu Feng, Lixia Yang, Lihong Chang, Xiaolong Zhu We consider a coupled system for image denoising, which consists of an anisotropic diffusion equation and a global sparse gradient model (GSG). The global sparse gradient model is used to estimate the edge map from the noisy image accurately and robustly. The anisotropic diffusion process is guided by the edge map. Furthermore, we prove the existence and uniqueness of the solutions of the coupled system with Dirichlet initial–boundary value problem. Finally, comparing with the edge indicators based on local methods, experimental results demonstrate that our proposed method has the state-of-the-art performance among the PDE-based anisotropic diffusion approaches both in objective measurement and visual evaluation.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Xi-Ming Fang, Zhi-Wei Zhu For large sparse linear complementarity problems, through reformulating them as implicit fixed-point equations, we propose a modulus-based matrix double splitting (MB-DS) iteration method by splitting the system matrices twice. Besides, the convergence of this method is proved when the system matrix is a P-matrix and an H+-matrix. In some special cases, we present the convergence regions and the optimal values for the parameter ω. In order to show the efficiency of the MB-DS iteration method and the effectiveness of the optimal parameter, some corresponding numerical experiments are performed.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Aly R. Seadawy, Mujahid Iqbal, Dianchen Lu In this research work, we constructed the solitary wave solutions of generalized Kadomtsev–Petviashvili modified equal width (KP-MEW) equation with the help of new technique which is modification form of extended auxiliary equation mapping method. The generalized KP-MEW equation is the nonlinear PDEs which described the propagation of long-wave with dissipation and dispersion in nonlinear media. As a result, families of solitary wave solutions are obtained in different form of solitons, bright–dark solitons and traveling wave solutions. The physical structure of these new solutions is shown graphically in two and three dimensions with the aid of computer software Mathematica. These obtained new solutions show the power and effectiveness of this new method. We can also solve other nonlinear system of PDEs which are involved in mathematical physics and many other branches of physical sciences with the help of this new method.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Luca Heltai, Nella Rotundo When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Xiao-Wei Gao, Hua-Yu Liu, Jun Lv, Miao Cui A novel strong form numerical method, Element Differential Method (EDM), is developed to solve geometrically complex mechanics problems based on triangular or tetrahedral meshes. The discretization of the structure under investigation has been based on Lagrange isoparametric quadrilateral or hexahedral elements while applying EDM. In this paper, a new family of isoparametric triangular and tetrahedral elements with a central node is proposed for EDM. A set of shape functions with analytical expressions for their first and second order partial derivatives is constructed for these triangular and tetrahedral elements, respectively. Moreover, a new element collocation scheme is proposed to establish a system of equations directly from the governing differential equations for internal nodes and traction-equilibrium equations for nodes on edges of an element. In this collocation scheme, no variational principles or virtual energy principles are required to set up the solution scheme, while no integration is needed when forming the coefficients of the system of equations. Numerical examples including standard patch tests and more practical problems are given to demonstrate the correctness of the constructed elements and the efficiency of the proposed element collocation method.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Lin Qiu, Fajie Wang, Ji Lin The transient heat conduction problems in layered materials are common encountered in practical application, but bring great challenge to most numerical algorithms. In this study, the singular boundary method (SBM), a meshless boundary collocation technique, is first employed to numerically simulate such problems based on the time-dependent fundamental solution of diffusion equation. Taking the boundary conditions and interface conditions into account, the computing system of the SBM for layered materials is established, and then is solved. The proposed method fully inherits the merits of conventional boundary-type methods while possessing its distinctive advantages. Furthermore, it is simple, straightforward, computationally efficient, and stable since it does not need to discretize temporal derivative term. Three numerical experiments are performed to verify the efficiency and accuracy of the proposed scheme. Numerical results clearly indicate the efficiency, accuracy and stability of the presented SBM for solving transient heat conduction problems in layered materials.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Mohammad Hemami, Kourosh Parand, Jamal Amani Rad Cognitive disorders especially epilepsy are closely linked with synchronization/desynchronization of neurons in the brain. In this paper, the dynamical modeling and behavior analysis of a FitzHugh–Nagumo neuron and also synchronization control of a network of FitzHugh–Nagumo neurons which promise the understanding of cognitive processing, are studied. To numerically simulate these models and in order to overcome their difficulties such as computational complexity, multi-dimensionality, non-linearity, having large spatial and temporal domains and also having a discrete initial data, we propose the use of a strongly meshless technique based on Wendland compactly supported radial basis function in conjunction with an operator splitting algorithm. The main advantage of the proposed algorithm is its computational complexity (for example O(Nx13+Nx23) of the inversion of all matrices) which is much lower than the complexity of pure radial basis functions method (O(Nx13Nx23) of the inversion of matrices). Numerical experiments are presented showing that the proposed approaches are extremely accurate and fast.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Andriy Beshley, Roman Chapko, B. Tomas Johansson A numerical implementation of the alternating iterative method is presented for the Cauchy problem of the conductivity equation in planar annular domains. The conductivity is space dependent as opposed to earlier works, where it is assumed constant. In the constant conductivity case, the mixed problems in the iterations are solved efficiently using boundary integrals. Following on to this, for a space dependent conductivity, it is outlined how to solve these mixed problems using a recent boundary-domain integral equations approach involving the parametrix. For a mixed problem the solution is written as a combination of boundary integrals and a domain integral, with densities to be determined. The densities needed over the boundary and domain are identified by matching against the given mixed boundary data and by requiring the governing equation to hold in the domain. An efficient Nyström scheme is applied for the discretisation. Numerical results are presented for several domains and conductivities, using exact as well as noisy Cauchy data, showing that a stable solution can be obtained with good accuracy and small computational cost.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Jaroslav Vondřejc This paper is focused on the double-grid integration with interpolation-projection (DoGIP), which is a novel matrix-free discretisation method of variational formulations introduced for Fourier–Galerkin approximation. Here, it is described as a more general approach with an application to the finite element method (FEM) on simplexes. The approach is based on treating the trial and a test function in variational formulation together, which leads to the decomposition of a linear system into interpolation and (block) diagonal matrices. It usually leads to reduced memory demands, especially for higher-order basis functions, but with higher computational requirements. The numerical examples are studied here for two variational formulations: weighted projection and scalar elliptic problem modelling, e.g. diffusion or stationary heat transfer. This paper also opens a room for further investigation, which is discussed in the conclusion.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Linlin Bu, Liquan Mei, Yan Hou In this paper, we propose stable second-order numerical schemes for the fractional Cahn–Hilliard and Allen–Cahn equations, which are based on the convex splitting in time and the Fourier spectral method in space. It is shown that the scheme for the fractional Cahn–Hilliard equation preserves mass. Meanwhile, the unique solvability and energy stability of the numerical schemes for the fractional Cahn–Hilliard and Allen–Cahn equations are proved. Finally, we present some numerical experiments to confirm the accuracy and the effectiveness of the proposed methods.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Xiaoqiang Yue, Shi Shu, Xiaowen Xu, Weiping Bu, Kejia Pan The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions, which is discretized by the space–time finite element method to propagate solutions. We develop a multigrid-reduction-in-time (MGRIT) algorithm with time-dependent time-grid propagators and provide its two-level convergence theory under the assumptions of the stability and simultaneous diagonalizability on time-grid propagators. Numerical results show that the proposed method possesses the saturation error order, theoretical results of the two-level variant deliver good predictions for our model problems, and significant speedups of the MGRIT can be achieved when compared to the two-level variant with F-relaxation (an equivalent version of the parareal algorithm) and the sequential time-stepping approach.

Abstract: Publication date: 1 December 2019Source: Computers & Mathematics with Applications, Volume 78, Issue 11Author(s): Ri Jin, Guirong Weng Distance regularized level set evolution (DRLSE) model, which solves the re-initialization problem in early active contours, is a ground breaking edge-based model for image segmentation. However, it has the disadvantages of unsatisfactory robustness to initialization and noise, unidirectional movement, slow convergence and poor stability. In this paper, we propose an active contour model driven by improved fuzzy c-means algorithm (FCM) and adaptive functions. An adaptive sign function based on image clustering information not only increases stability, but also solves the problem of unidirectional movement. Furthermore, it gives our model the ability to selectively segment targets in image. An adaptive edge indicator function accelerates convergence with better function performance. To further increase stability, a novel double-well potential function and the corresponding evolution speed function are proposed. Due to the improved FCM, the proposed model is robust to initialization and noise. In addition, our model exhibits an edge-based and region-based characteristic. Experimental results have proved that the proposed model can not only effectively segment images with intensity inhomogeneity, but also show a good robustness to initialization. Moreover, it has shorter time spent and higher segmentation accuracy compared with other models.

Abstract: Publication date: Available online 24 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Xiaolei Yuan, Zhenhua Chai, Huili Wang, Baochang Shi In this paper, a generalized lattice Boltzmann (LB) model with a source term in the continuity equation is proposed to solve both incompressible and nearly incompressible Navier–Stokes (N–S) equations. This model can be used to deal with single-phase and two-phase flows problems with a source term in the continuity equation. From this generalized model, we can not only get some existing models, but also derive new models. Moreover, for the incompressible model derived, a modified pressure scheme is introduced to calculate the pressure, and then to ensure the accuracy of the model. In this work, we will focus on a two-phase flow system, and in the frame work of our generalized LB model, a new phase-field-based LB model is developed for incompressible and quasi-incompressible two-phase flows. A series of numerical simulations of some classic physical problems, including a spinodal decomposition, a static droplet, a layered Poiseuille flow, and a bubble rising flow under buoyancy, are performed to validate the developed model. Besides, some comparisons with previous quasi-incompressible and incompressible LB models are also carried out, and the results show that the present model is accurate in the study of two-phase flows. Finally, we also conduct a comparison between quasi-incompressible and incompressible LB models for two-phase flow problems, and find that in some cases, the proposed quasi-incompressible LB model performs better than incompressible LB models.

Abstract: Publication date: Available online 24 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Hui Jin, Huibo Wang, Zhenqun Wu, Zhiwei Ge, Yunan Chen Supercritical water fluidized bed is a novel gasification reactor which can achieve efficient and clean utilization of coal. The rough surface of particle produced during grinding and thermochemical conversion processing will deeply affect supercritical water-particle two-phase flow and heat transfer characteristics. In this paper, fully resolved numerical simulation of supercritical water flow past single rough sphere particle with the Reynolds number ranging from 10 to 200 was carried out to investigate the effect of surface roughness. The simulation results show that as roughness increases, the separation bubbles generated in the dimple enhance the flow separation but has no significant effect on the drag coefficient. Particle surface-average Nusselt number decreases with an increase of roughness and surface enlargement coefficient due to the isolation effect at low Re and local separation bubbles in the dimple at high Re. Furthermore, the effect of surface enlargement coefficient on heat transfer efficiency factor for supercritical water near the critical point is greater than that under constant property condition and has a higher dependence on Re.

Abstract: Publication date: Available online 24 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Li-Min Liu, Ying-Ying Cui, Jie Xu, Chao Li, Qing-Hui Gao In this paper, the q-Gaussian process based on the non-extensive theory is discussed from a mathematical point of view, which has been widely applied to many anomalous diffusion systems in physics and finance. Firstly, the discussion of non-Markovian property of q-Gaussian process provides a numerical support for the future theoretical research. Secondly, the martingale and self-similarity of this process are obtained by Tsallis distributions. Thirdly, the long dependence is analyzed by simulations and Hurst exponents are compared with those of fractional Brownian motion. At last, the European call option price formula driven by this process is simulated, by which we find that this process can better match anomalous diffusion and the volatility smile.

Abstract: Publication date: Available online 23 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Sergiy Reutskiy, Ji Lin We present a RBF-based semi-analytical technique for solving 3D convection–diffusion–reaction (CDR) equations to model transport in an anisotropic inhomogeneous medium. The mathematical model is expressed as the boundary value problem for elliptic partial differential equation (EPDE). Main feature of the presented technique is the separately satisfaction of the conditions on the boundary of the domain and the EPDE inside. To be more precise, we transform the original EPDE to the equation with homogeneous boundary condition (BC) and seek the approximate solution as a sum of the modified RBFs (MRBFs). The MRBFs satisfy the homogeneous BC of the problem. So, any linear combination also satisfies the homogeneous BC. The RBFs of three types are used in the framework of the method: the Multiquadric (MQ) RBF, the Gaussian RBF and the conical one. The coefficients of the linear combination are determined so that it satisfies the governing equation of the EPDE. Ten numerical examples demonstrate the high effectiveness of the presented technique in solving 3D CDR problems in single and double connected domains.

Abstract: Publication date: Available online 23 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Bo An, J.M. Bergadà, F. Mellibovsky, W.M. Sang In order to study the flow behavior at high Reynolds numbers, two modified models, known as the multiple-relaxation-time lattice Boltzmann method (MRT-LBM) and large-eddy-simulation lattice Boltzmann method (LES-LBM), have been employed in this paper. The MRT-LBM was designed to improve numerical stability at high Reynolds numbers, by introducing multiple relaxation time terms, which consider the variations of density, energy, momentum, energy flux and viscous stress tensor. As a result, MRT-LBM is capable of dealing with turbulent flows considering energy dispersion and dissipation. In the present paper, this model was employed to simulate the flow at turbulent Reynolds numbers in wall-driven cavities. Two-sided wall driven cavity flow was studied for the first time, based on MRT-LBM, at Reynolds numbers ranging from 2×104to1×106, and employing a very large resolution2048 × 2048. It is found that whenever top and bottom lids are moving in the opposite directions, and the Reynolds number is higher than 2×104, the flow is chaotic, although some quasi-symmetric properties still remain, fully disappearing at Reynolds numbers between 2×105 and 3×105. Furthermore, between this Reynolds numbers range, 2×105

Abstract: Publication date: Available online 22 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Wei Liu, Huoyuan Duan In this paper, a virtual element method for the stochastic Stokes equations driven by an additive white noise is proposed and analyzed. The velocity is approximated by the lowest-order virtual element which is originally designed for the Poisson equation and the projection is also taken as the one originally for the Poisson equation, while the pressure is approximated by the traditional discontinuous piecewise constant element. For stable approximations, we adopt a stabilization associating with the pressure jumps. We show the inf-sup condition and derive the stability. We moreover obtain the error estimates in various norms and the estimates of the expectation of the errors through the Green function. Numerical results on polygonal mesh are presented to illustrate the performance of the proposed method and the theoretical results obtained.

Abstract: Publication date: Available online 19 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Mostafa Abaali, Zoubida Mghazli In the biological units in situ, there is proved the existence of free and adherent bacteria. Most mathematical models deal with only free or adherent bacteria, and do not consider the bacteria diffusion and transport. In this work we consider a mathematical model of biodenitrifcation taking into account free and adherent bacteria, their interdependence and the diffusion and transport related to them. The model is presented first in a reactor and then in a porous medium. The equations obtained are approximated by a Finite Element method. The numerical tests presented are close to experimental results in the literature, that confirm the validity of the model, and show the importance of considering the complete model taking into account both types of bacteria and their evolution in time and space.

Abstract: Publication date: Available online 18 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Owe Axelsson A preconditioned square block matrix, called PRESB has previously been applied successfully and, for more standard type of problems, have been shown to have eigenvalue bounds in the interval (1∕2,1], which holds uniformly with respect to all parameters involved. Having such fixed bounds enables the use of an inner-product free acceleration method, such as the Chebyshev iterative method. Here it is shown that the method can be applied also for some more general problems, where the spectrum contains outlier eigenvalues larger than unity and that one can apply a polynomially modified version of the Chebyshev method for such problems.

Abstract: Publication date: Available online 18 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Hanquan Wang, Ronghua Cheng, Xinming Wu In this paper, we propose a splitting Fourier pseudospectral method for Vlasov–Poisson–Fokker–Planck (VPFP) system, which describes the motion of charged particles in plasma. The numerical integration for the system is performed by employing the splitting method in time, Fourier Galerkin method in space direction, and Fourier collocation method in phase direction, respectively. The algorithm has spectral accuracy in both space and phase directions and can be implemented efficiently with the fast Fourier transform and technique of diagonalization, respectively. Extensive numerical results in one-dimensional phase space (or 1x×1v) from the proposed numerical method are shown and have proven the good agreement with the theory and previous studies. Numerical algorithm for the VPFP system in two-dimensional phase space (or 2x×2v) has been summarized in Appendix.

Abstract: Publication date: Available online 18 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Rooholah Abedian, Rezvan Salehi The aim of this paper is to study the numerical application of radial basis functions (RBFs) approximation in the reconstruction process of well known ENO/WENO schemes. The resulted schemes are employed for approximating the viscosity solution of Hamilton–Jacobi (H–J) equations. The accuracy in the smooth area is enhanced by locally optimizing the shape parameter according to the results. It is revealed that the proposed schemes in this research prepare more accurate reconstructions and sharper solution near singularities by comparing the RBFENO/RBFWENO schemes and the classical ENO/WENO schemes for some benchmark examples. Looking at the several numerical examples in 1D, 2D and 3D illustrate that the proposed schemes in this paper perform better than the traditional ENO/WENO schemes for solving H–J equations.

Abstract: Publication date: Available online 10 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Tao Liu In the present contribution, we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems, in which the forward problems are modeled by some forms of partial differential equations. The method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub-inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest, and then carries out the inversion on the coarsest grid by the homotopy method. The strategy may give a rapidly and globally convergent method. As a practical application, this method is used to solve the nonlinear inverse problem of a nonlinear convection–diffusion equation, which is the saturation equation within the two-phase porous media flow. We demonstrate the effectiveness and merits of the multigrid–homotopy method on two actual model problems.

Abstract: Publication date: Available online 10 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Ali Hashemian, Esmail Lakzian, Amir Ebrahimi-Fizik The isogeometric finite volume analysis is utilized in this research to numerically simulate the two-dimensional viscous wet-steam flow between stationary cascades of a steam turbine for the first time. In this approach, the analysis-suitable computational mesh with “curved” boundaries is generated for the fluid flow by employing a non-uniform rational B-spline (NURBS) surface that describes the cascade geometry, and the governing equations are then discretized by the NURBS representation. Thanks to smooth and accurate geometry representation of the NURBS formulation, the employed isogeometric framework not only resolves issues concerning the conventional mesh generation techniques of the finite volume method in steam turbine problems, but also, as validated against well-established experiments, significantly improves the accuracy of the numerical solution. In addition, the shock location in the cascade is predicted and tracked with a sufficient accuracy.

Abstract: Publication date: Available online 9 October 2019Source: Computers & Mathematics with ApplicationsAuthor(s): M.E. Hubbard, M. Ricchiuto, D. Sármány This article investigates the potential for an r-adaptation algorithm to improve the efficiency of space–time residual distribution schemes in the approximation of time-dependent hyperbolic conservation laws, e.g. scalar advection, shallow water flows, on unstructured, triangular meshes. In this adaptive framework the connectivity of the mesh, and hence the number of degrees of freedom, remain fixed, but the mesh nodes are continually “relocated” as the flow evolves so that features of interest remain resolved as they move within the domain.Adaptive strategies of this type are well suited to the space–time residual distribution framework because, when the discrete representation is allowed to be discontinuous in time, these algorithms can be designed to be positive (and hence stable) for any choice of time-step, even on the distorted space–time prisms which arise from moving the nodes of an unstructured triangular mesh. Consequently, a local increase in mesh resolution does not impose a more restrictive stability constraint on the time-step, which can instead be chosen according to accuracy requirements. The order of accuracy of the fixed-mesh scheme is retained on the moving mesh in the majority of applications tested.Space–time schemes of this type are analogous to conservative ALE formulations and automatically satisfy a discrete geometric conservation law, so moving the mesh does not artificially change the flow volume for pure conservation laws. For shallow water flows over variable bed topography, the so-called C-property (retention of hydrostatic balance between flux and source terms, required to maintain the steady state of still, flat, water) can also be satisfied by considering the mass balance equation in terms of free surface level instead of water depth, even when the mesh is moved.The r-adaptation is applied within each time-step by interleaving the iterations of the nonlinear solver with updates to mesh node positions. The node movement is driven by a monitor function based on weighted approximations of the scaled gradient and Laplacian of the local solution and regularised by a smoothing iteration. Numerical results are shown in two dimensions for both scalar advection and for shallow water flow over a variable bed which show that, even for this simple implementation of the mesh movement, reductions in cpu times of up to 60% can be attained without increasing the error.

Abstract: Publication date: Available online 24 September 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Alexey Chernov, Anne Reinarz The aim of this paper is to develop and analyse stable and accurate numerical approximation schemes for boundary integral formulations of the heat equation with Dirichlet boundary conditions. The accuracy of Galerkin discretisations for the resulting boundary integral formulations strongly depends on the choice of discretisation space. We develop a-priori error analysis utilising a proof technique that involves norm bounds in hierarchical wavelet subspace decompositions. We apply this to full tensor product discretisations and anisotropic sparse grid discretisations and demonstrate improvements over existing results in both cases. Finally, a simple adaptive scheme is proposed to suggest an optimal shape for the sparse grid index sets.