Abstract: Publication date: Available online 19 May 2017 Source:Computers & Mathematics with Applications Author(s): Gil Ho Yoon This study considers failure theories for brittle and ductile materials in the stress-based topology optimization method (STOM) for steady state fluid–structure interactions (FSI). In some relevant studies, the subject of the stress-based topology optimization to minimize volumes with local von Mises stress constraints has been researched. However, the various failure theories for ductile and brittle materials, such as the maximum shear stress theory, the brittle and ductile Mohr–Coulomb theory, and the Drucker–Prager theory, have not been considered. For successful STOM for FSI, in addition to alleviating physics interpolation issues between structure and fluid and some numerical issues related to STOM, the mathematical characteristics of the various failure theories should be properly formulated and constrained. To resolve all the involved computational issues, the present study applies the monolithic analysis method, the qp-relaxation method, and the p -norm approach to the failure constraints. The present topology optimization method can create optimal layouts while minimizing volume constraining local failure constraints for ductile and brittle materials for steady state fluid and structural interaction system.

Abstract: Publication date: Available online 19 May 2017 Source:Computers & Mathematics with Applications Author(s): Zhoufeng Wang, Yunzhang Zhang Maxwell’s equations for conical diffraction can be reduced to a system of two Helmholtz equations in R 2 coupled via quasi-periodic transmission conditions on a set of piecewise smooth interfaces. A finite element formulation of the conical diffraction problem is presented in a bounded domain by introducing the nonlocal boundary operators. An a posteriori error estimate is established when the truncation of the nonlocal boundary operators takes place. As our work in Wang et al. (2015), a duality argument is applied to overcome the difficulty caused by the fact that the truncated pseudo-differential mapping does not converge to the original pseudo-differential mapping in its operator norm. The a posteriori error estimate consists of two parts: finite element discretization error and the truncation error of the nonlocal boundary operators. In particular, the truncation error is exponentially decaying with respect to the truncation parameter.

Abstract: Publication date: Available online 17 May 2017 Source:Computers & Mathematics with Applications Author(s): Siwei Duo, Lili Ju, Yanzhi Zhang In this paper, we propose a fast algorithm for efficient and accurate solution of the space–time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian ( − Δ ) s α / 2 results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix–vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method.

Abstract: Publication date: Available online 17 May 2017 Source:Computers & Mathematics with Applications Author(s): Rastin Matin, Marek Krzysztof Misztal, Anier Hernández-García, Joachim Mathiesen In contrast to the commonly used lattice Boltzmann method, off-lattice Boltzmann methods decouple the velocity discretization from the underlying spatial grid thereby potentially increasing the geometric flexibility of the method. The current work combines characteristic-based integration of the streaming step with the free-energy based multiphase model by Lee and Lin (2005). This allows for simulation time steps more than an order of magnitude larger than the relaxation time. Unlike previous work by Wardle and Lee (2013) that integrated intermolecular forcing terms in the advection term, the current scheme applies collision and forcing terms locally for a simpler finite element formulation. A series of thorough benchmark studies reveal that this does not compromise stability and that the scheme is able to accurately simulate flows at large density and viscosity contrasts.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): A. Gil, J.P.G. Galache, C. Godenschwager, U. Rüde Simulations of the flow field through chaotic porous media are powerful numerical challenges of special interest in science and technology. The simulations are usually done over representative samples which summarise the properties of the material. Several factors affect the accuracy of the results. Firstly the spatial resolution has to be fine enough to be able to capture the smallest geometrical details. Secondly the domain size has to be large enough to contain the large characteristic scale of the porous media. And finally the effects induced by the boundary conditions have to be diluted when more realistic options are not available. This is the case when the geometry is obtained by tomography and the periodic boundary conditions cannot be applied to delimit the sample because its geometry is not periodic. Impermeable boundary conditions are usually chosen to enclose the domain, forcing mass conservation. As a result, the flow field is over-restricted and the total pressure drop can be over-estimated. In this paper a new strategy is presented to optimise the computational resources consumption keeping the restrictions imposed by the accuracy criteria. The effects of the domain size, discretisation thickness and boundary condition disturbances are studied in detail. The study starts with the procedural generation of chaotic porous walls which mimics acicular mullite filters. An advantage of this process is the possibility to create periodic geometries. Periodicity permits the application of advanced techniques such as cyclic cross-correlations between the phase field and the velocity component fields without aliasing. From cross-correlation operations the large characteristic scale is obtained. The result is a lower threshold for the domain size. In second place a mesh independent study is done to find the upper threshold for the lattice spacing. The Minkowski–Bouligand fractal dimension of the fluid–solid interface corroborates the results. It has been demonstrated how the fractal dimension is a good candidate to replace the mesh independent study with lower computational cost for this type of problems. The last step is to compare the results obtained for a periodic geometry applying periodicity and symmetry as boundary conditions. Considering the periodic case as reference the resultant error is analysed. The explanation of the analysis includes how the intensity of the error changes in space and the limitations of symmetric boundary conditions.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Huamin Zhang, Hongcai Yin This paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations ∑ j = 1 q A i j X j B i j = F i , i = 1 , 2 , … , p . We prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors. Also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum-norm solution.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Salman A. Malik, Sara Aziz We consider the inverse problem of determination of the solution and a source term for a time fractional diffusion equation in two dimensional space. The time fractional derivative is the Hilfer derivative. A bi-orthogonal system of functions in L 2 ( Ω ) , obtained from the associated non-self-adjoint spectral problem and its adjoint problem, is used to prove the existence and uniqueness of the solution of the inverse problem. The stability of the solution of the inverse problem on the given data is proved.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Lyubomir Boyadjiev, Yuri Luchko In this paper, a multi-dimensional α -fractional diffusion–wave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order α / 2 and the Riesz space-fractional derivative of the order α so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the α -fractional diffusion–wave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the α -fractional diffusion–wave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Zujin Zhang, Zheng-an Yao This paper concerns with the regularity criteria for the 3 D axisymmetric MHD system. It is proved that the control of swirl component of vorticity can ensure the smoothness of the solution.

Abstract: Publication date: 15 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 12 Author(s): Hong-Ru Xu, Shui-Lian Xie In this paper, we present a semismooth Newton method for a kind of HJB equation. By suitably choosing the initial iterative point, the method is proved to have monotone convergence. Moreover, the semismooth Newton method has local superlinear convergence rate. Some simple numerical results are reported.

Abstract: Publication date: Available online 13 May 2017 Source:Computers & Mathematics with Applications Author(s): Sitong Chen, Xianhua Tang This paper is dedicated to studying the following Schrödinger–Poisson system { − △ u + V ( x ) u + ϕ u = K ( x ) f ( u ) , x ∈ R 3 , − △ ϕ = u 2 , x ∈ R 3 , where V , K ∈ C ( R 3 , R ) and f ∈ C ( R , R ) . Under mild assumptions on the decay rate of V ( x ) , we establish the existence of one ground state sign-changing solution with precisely two nodal domains, by using a weaker growth condition lim ∣ t ∣ → ∞ ∫ 0 t f ( s ) d s ∣ t ∣ 3 = ∞ , instead of the usual super-cubic condition lim ∣ t ∣ → ∞ ∫ 0 t f ( s ) d s ∣ t ∣ 4 = ∞ . Our result unifies both asymptotically cubic and super-cubic cases, which generalizes and improves the corresponding ones in the literatures.

Abstract: Publication date: Available online 13 May 2017 Source:Computers & Mathematics with Applications Author(s): Giuseppe Balduzzi, Simone Morganti, Ferdinando Auricchio, Alessandro Reali The present paper combines an effective beam theory with a simple and accurate numerical technique opening the door to the prediction of the structural behavior of planar beams characterized by a continuous variation of the cross-section geometry, that in general deeply influences the stress distribution and, therefore, leads to non-trivial constitutive relations. Accounting for these peculiar aspects, the beam theory is described by a mixed formulation of the problem represented by six linear Ordinary Differential Equations (ODEs) with non-constant coefficients depending on both the cross-section displacements and the internal forces. Due to the ODEs’ complexity, the solution can be typically computed only numerically also for relatively simple geometries, loads, and boundary conditions; however, the use of classical numerical tools for this problem, like a (six-field) mixed finite element approach, might entail several issues (e.g., shear locking, ill-conditioned matrices, etc.). Conversely, the recently proposed isogeometric collocation method, consisting of the direct discretization of the ODEs in strong form and using the higher-continuity properties typical of spline shape functions, allows an equal order approximation of all unknown fields, without affecting the stability of the solution. This makes such an approach simple, robust, efficient, and particularly suitable for solving the system of ODEs governing the non-prismatic beam problem. Several numerical experiments confirm that the proposed mixed isogeometric collocation method is actually cost-effective and able to attain high accuracy.

Abstract: Publication date: Available online 12 May 2017 Source:Computers & Mathematics with Applications Author(s): José L. Galán-García, Gabriel Aguilera-Venegas

Abstract: Publication date: Available online 11 May 2017 Source:Computers & Mathematics with Applications Author(s): Ricardo Castillo, Miguel Loayza We are concerned with the parabolic equation u t − Δ u = f ( t ) u p ( x ) in Ω × ( 0 , T ) with homogeneous Dirichlet boundary condition, p ∈ C ( Ω ) , f ∈ C ( [ 0 , ∞ ) ) and Ω is either a bounded or an unbounded domain. The initial data is considered in the space { u 0 ∈ C 0 ( Ω ) ; u 0 ≥ 0 } . We find conditions that guarantee the global existence and the blow up in finite time of nonnegative solutions. These conditions are given in terms of the asymptotic behavior of the solution of the homogeneous linear problem u t − Δ u = 0 .

Abstract: Publication date: Available online 11 May 2017 Source:Computers & Mathematics with Applications Author(s): Andrzej Karafiat A classical mixed boundary-value problem of linear elasticity in two dimensions in the Galerkin boundary integral formulation is considered. We prove a-priori error estimates of the solution to this problem by the isogeometric adaptive method using NURBS. The estimates include approximation of the boundary of the domain.

Abstract: Publication date: Available online 11 May 2017 Source:Computers & Mathematics with Applications Author(s): Jianping Yu, Yongli Sun In this paper, we study the Gaussian solitary waves for some nonlinear evolution equations with logarithmic nonlinearities. These studied logarithmic evolution equations are the generalized logarithmic BBM equations, the logarithmic ( 2 + 1 ) -dimensional KP-like equations, the logarithmic ( 3 + 1 ) -dimensional KP-like equations, the generalized logarithmic ( 2 + 1 ) -dimensional Klein-Gordon equations and the generalized logarithmic ( 3 + 1 ) -dimensional Klein-Gordon equations. We not only prove that they possess Gaussons: solitary wave solutions of Gaussian shape but also derive the relationships among the parameters.

Abstract: Publication date: Available online 11 May 2017 Source:Computers & Mathematics with Applications Author(s): Ola Ragb, L.F. Seddek, M.S. Matbuly A numerical scheme based on differential quadrature methods, is introduced for solving Bratu problem. The problem is firstly reduced to an iterative one. Then, both of differential quadrature method (DQM) and moving least squares differential quadrature method (MLSDQM) are applied to solve iteratively the nonlinear problem. The proposed scheme successfully computes multiple solutions to Bratu’s problem. The obtained results agree with the 1D and 2D closed forms. Further a parametric study is introduced to investigate the computational characteristics of the proposed scheme.

Abstract: Publication date: 1 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 11 Author(s): Xi Yang We consider a waveform relaxation (WR) method based on the Hermitian/skew-Hermitian splitting (HSS) of the system matrices, which is a continuous-time iteration method. In actual implementation, the continuous-time WR-HSS method (CWR-HSS) is replaced by the discrete-time WR-HSS method (DWR-HSS) defined on a time-level-sequence. If the time-step-size tends to zero, the approximate solution obtained by the DWR-HSS method on each time level is proved to converge to the limit of the approximate solution obtained by the CWR-HSS method, i.e., the exact solution of a corresponding system of linear differential equations. Finally, the above relationship between the CWR-HSS method and the DWR-HSS method is verified by the numerical tests based on the unsteady discrete elliptic problem. Therefore, the DWR-HSS method is a reliable option for solving the unsteady discrete elliptic problem in both theoretical and practical aspects.

Abstract: Publication date: 1 June 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 11 Author(s): Li-Dan Liao, Guo-Feng Zhang For fast solving weighted Toeplitz least-squares problems from image restoration, Ng and Pan (2014) studied a new Hermitian and skew-Hermitian splitting (NHSS) preconditioner. In this paper, a generalization of the NHSS preconditioner and the corresponding iterative method are presented. Convergence for the new iteration method is studied and optimal choice of the parameters is discussed. Bounds on the eigenvalues and the corresponding eigenvector distributions are proposed. The degree of the minimal polynomial of the preconditioned matrix is obtained. Numerical experiments arising from image restoration are provided, which show the proposed iteration method is effective and confirm our theoretical results are correct.

Abstract: Publication date: Available online 9 May 2017 Source:Computers & Mathematics with Applications Author(s): Shaolei Ru, Jiecheng Chen In this paper, we first prove the local well-posedness of the fractional Burgers equations in N Dimensions. Combining the local well-posedness and the method of modulus of continuity, we show the global well-posedness of the N-D critical Burgers equation in critical Besov spaces B ̇ p , 1 N p ( R N ) with p ∈ [ N , ∞ ) .

Abstract: Publication date: Available online 8 May 2017 Source:Computers & Mathematics with Applications Author(s): Hongyu Ye In this paper, we study the existence and the concentration behavior of critical points for the following functional derived from the Schrödinger–Poisson system: E ( u ) = 1 2 ∫ R 3 ∇ u 2 + 1 4 ∫ R 3 ( x − 1 ∗ u 2 ) u 2 − 3 10 ∫ R 3 u 10 3 constrained on the L 2 -spheres S ( c ) = { u ∈ H 1 ( R 3 ) u 2 = c } when c > c ∗ = Q 2 , where Q is up to translations, the unique positive of − Δ Q + 2 3 Q = Q 4 3 Q in R 3 . As such constrained problem is L 2 -critical, E ( u ) is unbounded from below on S ( c ) when c > c ∗ and the existence of critical points constrained on S ( c ) is obtained by a mountain pass argument on S ( c ) . We show that there exists c 1 > ( 9 7 ) 3 4 c ∗ such that E ( u ) has at least one positive critical point restricted to S ( c ) for c ∗ < c ≤ c 1 . As c approaches c PubDate: 2017-05-11T13:24:22Z

Abstract: Publication date: Available online 8 May 2017 Source:Computers & Mathematics with Applications Author(s): M. Faustmann, J.M. Melenk The h p -version of the finite element method is applied to singularly perturbed reaction–diffusion type equations on polygonal domains. The solution exhibits boundary layers as well as corner layers. On a class of meshes that are suitably refined near the boundary and the corners, robust exponential convergence (in the polynomial degree) is shown in both a balanced norm and the maximum norm.

Abstract: Publication date: Available online 6 May 2017 Source:Computers & Mathematics with Applications Author(s): Hailong Yuan, Jianhua Wu, Yanling Li In this paper, a two-species cooperative model with diffusion and under homogeneous Dirichlet boundary conditions is investigated. It is shown the existence, stability, uniqueness and multiplicity of positive solutions. In particular, we study the global asymptotical stability of the unique positive solution when a ∈ ( λ 1 ( − b θ d 1 + β θ d ) , ∞ ) and α is large. Our method of analysis is based on perturbation technique, the Lyapunov–Schmidt procedure and the bifurcation theory.

Abstract: Publication date: Available online 5 May 2017 Source:Computers & Mathematics with Applications Author(s): Sashikumaar Ganesan, Shangerganesh Lingeshwaran A finite element scheme for the solution of a cancer invasion model is proposed. The cancer dynamics model consists of three coupled partial differential equations which describe the evolution of cancer cell density, extra cellular matrix and the matrix degrading enzymes. The model incorporates proliferation and haptotaxis effect of cancer cells, their interaction with extracellular matrix, the production of matrix degrading enzymes and consequent degradation of the extracellular matrix. The coupled partial differential equations are discretized in space with the standard Galerkin finite elements and in time with the Crank–Nicolson method. Moreover, the nonlinear terms in the coupled equations are treated semi-implicitly in the finite element scheme. The numerical scheme is validated with numerical results taken from the literature. In addition to the mesh convergence study, the effects of haptotactic rate, proliferation rate and remodelling rate of matrix components of the considered mathematical model are investigated.

Abstract: Publication date: Available online 2 May 2017 Source:Computers & Mathematics with Applications Author(s): Xiazhi Hao, Yinping Liu, Xiaoyan Tang, Zhibin Li Through the truncated Painlevé expansion, the residual symmetry of the Davey–Stewartson III system is derived. This symmetry is localized to a properly prolonged system and the corresponding Bäcklund transformation is obtained. Based on the transformation, some exact solutions are generated. Furthermore, several types of interaction wave solutions are obtained by using the consistent tanh expansion method.

Abstract: Publication date: Available online 2 May 2017 Source:Computers & Mathematics with Applications Author(s): Marc Bakry, Sébastien Pernet, Francis Collino In this work we construct a new reliable, efficient and local a posteriori error estimate for the single layer and hyper-singular boundary integral equations associated to the Helmholtz equation in two dimensions. It uses a localization technique based on a generic operator Λ which is used to transport the residual into L 2 . Under appropriate conditions on the construction of Λ , we show that it is asymptotically exact with respect to the energy norm of the error. The single layer equation and the hyper-singular equation are treated separately. While the current analysis requires the boundary to be smooth, numerical experiments show that the new error estimators also perform well for non-smooth boundaries.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): Xiaoli Li, Hongxing Rui In this paper, the characteristic block-centered finite difference method is introduced and analyzed to solve the incompressible wormhole propagation. Error estimates for the pressure, velocity, porosity, concentration and its flux in different discrete norms are established rigorously and carefully on non-uniform grids. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): Hailong Qiu, Yongchao Zhang, Liquan Mei In this work we consider a mixed finite element method (FEM) for incompressible Navier–Stokes equations with nonlinear damping term and friction boundary conditions. After establishing the variational formulation, we show the well-posedness of this problem. Subsequently, we focus our attention on the mixed FEM and analyze the Galerkin approximation to this system. Then some optimal error estimates are deduced. In the end, some iterative algorithms are presented and numerical results are given to verify the theoretical analysis.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): Si-Tao Ling, Zhi-Gang Jia, Tong-Song Jiang The solution of a linear quaternionic least squares (QLS) problem can be transformed into that of a linear least squares (LS) problem with JRS-symmetric real coefficient matrix, which is suitable to be solved by developing structured iterative methods when the coefficient matrix is large and sparse. The main aim of this work is to construct a structured preconditioner to accelerate the LSQR convergence. The preconditioner is based on structure-preserving tridiagonalization to the real counterpart of the coefficient matrix of the normal equation, and the incomplete inverse upper–lower factorization related to only one symmetric positive definite tridiagonal matrix rather than four, so it is reliable and has low storage requirements. The performances of the LSQR algorithm with structured preconditioner are demonstrated by numerical experiments.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): Hongyan Xie, Mingxuan Zhu This paper proves a regularity criterion ∇ u , ∇ b ∈ L ∞ ( 0 , T ; L ∞ ) for 3D ideal density-dependent MHD system.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): D. Castaño, M.C. Navarro, H. Herrero In this paper we use nonlinear simulations to show the generation of single-cell vortices, two-cell vortices and double vortices in a cylinder non-homogeneously heated from below in a rotating reference frame. For moderate rotation rates we show the transition from an axisymmetric one-cell vortex, characterized by an updraft at the center of the cell, to an axisymmetric two-cell vortex, characterized by a central downdraft surrounded by updraft, i.e., a vortex that develops a central eye. This transition can be explained through a force balance analysis. When the thermal gradient increases beyond a certain threshold the axisymmetric single-eyed vortex loses the axisymmetry, the eye displaces from the center and tilts. For larger rotation rates the axisymmetric one-cell vortex bifurcates to a double vortex.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): Xiaoying Shen, Qiaozhen Ma In this paper, we prove the existence of random attractors for the continuous random dynamical systems generated by stochastic weakly dissipative plate equations with linear memory and additive white noise by defining the energy functionals and using the compactness translation theorem.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): Stefania Monica, Federico Bergenti This paper presents an analytic approach that can be used to study opinion dynamics in multi-agent systems. The results of such an analytic approach can be used as a descriptive tool capable of predicting the long-term properties of a multi-agent system, and they can also be considered a prescriptive tool that supports the design of multi-agent systems with desired asymptotic characteristics. The agents that form the multi-agent system are divided into disjoint classes characterized by different values of fixed parameters to account for the specific behaviors of single agents. Each class is characterized by the number of agents in it, by the initial distribution of the opinion, and by the characteristic propensity of single agents to change their respective opinions when interacting with other agents. The proposed approach is based on the possibility of interpreting the dynamics of the opinion in terms of the kinetic theory of gas mixtures, which allows expressing the dynamics of the average opinion of each class in terms of a suitable differential problem that can be used to derive interesting asymptotic properties. Analytic solutions of the obtained differential problem are derived and it is shown that, under suitable hypotheses, the average opinions of all classes of agents converge to the same value. The results presented in this paper differ from those commonly derived in standard kinetic theory of gas mixtures because the microscopic equations which describe the effects of interactions among agents are explicitly meant to model opinion dynamics, and they are different from those normally used to describe collisions among molecules in a gas. All presented analytic results are confirmed by simulations presented at the end of the paper.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): Mario Kapl, Vito Vitrih We study the space of C 2 -smooth isogeometric functions on bilinearly parameterized multi-patch domains Ω ⊂ R 2 , where the graph of each isogeometric function is a multi-patch spline surface of bidegree ( d , d ) , d ∈ { 5 , 6 } . The space is fully characterized by the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. Groisser and Peters (2015), Kapl et al. (2015)). This is the reason to call its functions C 2 -smooth geometrically continuous isogeometric functions. In particular, the dimension of this C 2 -smooth isogeometric space is investigated. The study is based on the decomposition of the space into three subspaces and is an extension of the work Kapl and Vitrih (2017) to the multi-patch case. In addition, we present an algorithm for the construction of a basis, and use the resulting globally C 2 -smooth functions for numerical experiments, such as performing L 2 approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order.

Abstract: Publication date: 15 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 10 Author(s): Hai-Qiang Zhang, Wen-Xiu Ma The linear superposition principle can apply to the construction of resonant multiple wave solutions for a ( 3 + 1 )-dimensional nonlinear evolution equation. Two types of resonant solutions are obtained by the parameterization for wave numbers and frequencies for linear combinations of exponential traveling waves. The resonance phenomena of multiple waves are discussed through the figures for several sample solutions.

Abstract: Publication date: Available online 29 April 2017 Source:Computers & Mathematics with Applications Author(s): Dongsheng Cheng, Xu Tan, Taishan Zeng In this paper, we develop a new dispersion minimizing finite difference scheme for the Helmholtz equation with perfectly matched layer (PML) in two dimensional domain, which is a second order 9-point scheme. To discretize the second derivative operator, we employ a linear combination of a point and its neighboring grid points to replace each of the five points in the traditional central difference scheme. Based on minimizing the numerical dispersion, the combination weights are determined by minimizing the numerical dispersion with a flexible selection strategy. The new scheme is simple, rotation-free, and pointwise consistent with the equation, which is different from the classical rotated 9-point difference scheme obtained by combining the Cartesian coordinate system and the rotated system. Moreover, it is a robust scheme even if the step sizes of different directions are not equal. Convergence analysis and dispersion analysis are given. Several numerical examples are presented to illustrate the numerical convergence and effectiveness of the new scheme.

Abstract: Publication date: Available online 29 April 2017 Source:Computers & Mathematics with Applications Author(s): Wen Li, Song Wang In this paper we propose a combination of a penalty method and a finite volume scheme for a four-dimensional time-dependent Hamilton–Jacobi–Bellman (HJB) equation arising from pricing European options with proportional transaction costs and stochastic volatility. The HJB equation is first approximated by a nonlinear partial differential equation containing penalty terms. A finite volume method along with an upwind technique is then developed for the spatial discretization of the nonlinear penalty equation. We show that the coefficient matrix of the discretized system is an M -matrix. An iterative method is proposed for solving the nonlinear algebraic system and a convergence theory is established for the iterative method. Numerical experiments are performed using a non-trivial model pricing problem and the numerical results demonstrate the usefulness of the proposed method.

Abstract: Publication date: Available online 28 April 2017 Source:Computers & Mathematics with Applications Author(s): Dongdong He, Kejia Pan In this paper, we propose a three-level linearly implicit combined compact difference method (CCD) together with alternating direction implicit method (ADI) for solving the generalized nonlinear Schrödinger equation (NLSE) with variable coefficients in two and three dimensions. The method is sixth-order accurate in space variable and second-order accurate in time variable. Fourier analysis shows that the method is unconditionally stable. Comparing to the nonlinear CCD–PRADI scheme for solving the 2D cubic NLSE with constant coefficients (Li et al., 2015), current method is a linear scheme which generally requires much less computational cost. Moreover, current method can handle 3D problems with variable coefficients naturally. Finally, numerical results for both 2D and 3D cases are presented to illustrate the advantages of the proposed method.

Abstract: Publication date: Available online 26 April 2017 Source:Computers & Mathematics with Applications Author(s): N. Sakib, A. Mohammadi, J.M. Floryan A three-dimensional, spectrally accurate algorithm based on the immersed boundary conditions (IBC) concept has been developed for the analysis of flows in channels bounded by rough boundaries. The algorithm is based on the velocity–vorticity formulation and uses a fixed computational domain with the flow domain immersed in its interior. The geometry of the boundaries is expressed in terms of double Fourier expansions and boundary conditions enter the algorithm in the form of constraints. The spatial discretization uses Fourier expansions in the stream-wise and span-wise directions and Chebyshev expansions in the wall-normal direction. The algorithm can use either the fixed-flow-rate constraint or the fixed-pressure-gradient constraint; a direct implementation of the former constraint is described. An efficient solver which takes advantage of the structure of the coefficient matrix has been developed. It is demonstrated that the applicability of the algorithm can be extended to more extreme geometries using the over-determined formulation. Various tests confirm the spectral accuracy of the algorithm.

Abstract: Publication date: Available online 25 April 2017 Source:Computers & Mathematics with Applications Author(s): Maciej Smołka In this paper we study the Fréchet differentiability of the objective functional for a quite general class of coefficient inverse problems. We present sufficient conditions for the existence of arbitrary-order differentials as well as formulae for the calculation of first and second order derivatives. The general case theorems are then applied to two real-world geophysical inverse problems.

Abstract: Publication date: Available online 20 April 2017 Source:Computers & Mathematics with Applications Author(s): Jaemin Shin, Hyun Geun Lee, June-Yub Lee In this paper, we present the Convex Splitting Runge–Kutta (CSRK) methods which provide a simple unified framework to solve phase-field models such as the Allen–Cahn, Cahn–Hilliard, and phase-field crystal equations. The core idea of the CSRK methods is the combination of convex splitting methods and multi-stage implicit–explicit Runge–Kutta methods. Our CSRK methods are high-order accurate in time and we investigate the energy stability numerically. We present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third-order accuracy.

Abstract: Publication date: Available online 19 April 2017 Source:Computers & Mathematics with Applications Author(s): A.Z. Fino, H. Ibrahim, A. Wehbe We consider the initial boundary value problem of the nonlinear damped wave equation in an exterior domain Ω . We prove a blow-up result which generalizes the result of non-existence of global solutions of Ogawa and Takeda (2009). We also show that the critical exponent belongs to the blow-up case.

Abstract: Publication date: Available online 18 April 2017 Source:Computers & Mathematics with Applications Author(s): Jishan Fan, Yong Zhou This paper proves some regularity criteria for the 3D (density-dependent) incompressible Maxwell–Navier–Stokes system, which improves a recent result of Kang and Lee (2013).

Abstract: Publication date: Available online 11 April 2017 Source:Computers & Mathematics with Applications Author(s): Lingyu Jin, Lang Li, Shaomei Fang We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space R n , n ≥ 1 . Here, the fractional order α is related to the diffusion-type source term behaving as the usual diffusion term on the high frequency part. It has a feature of regularity-gain and regularity-loss for α > 1 and 0 < α < 1 , respectively. We establish the global existence and time-decay rates for small-amplitude solutions to the Cauchy problem for α > 0 . In the case that 0 < α < 1 , we introduce the time-weighted energy method to overcome the weakly dissipative property of the equation.

Abstract: Publication date: Available online 9 April 2017 Source:Computers & Mathematics with Applications Author(s): Jing-Tao Li, Chang-Feng Ma In this paper, we propose the parameterized upper and lower triangular (denoted by PULT) splitting iteration methods for solving singular saddle point problems. The eigenvalues and eigenvectors of iteration matrix of the new methods are studied. It is shown that the proposed methods are semi-convergent under certain conditions. Besides, the pseudo-optimal iteration parameter and corresponding convergence factor can be obtained in some special cases of the PULT iteration methods. Numerical example is presented to confirm the theoretical results, which implies that PULT iteration methods are effective and feasible for solving singular saddle point problems.

Abstract: Publication date: Available online 9 April 2017 Source:Computers & Mathematics with Applications Author(s): Marcelo M. Cavalcanti, Wellington J. Corrêa, Carole Rosier, Flávio R. Dias Silva In this paper, we obtain very general decay rate estimates associated to a wave–wave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in Lasiecka and Tataru (1993) and we shall present explicit decay rate estimates as considered in Alabau-Boussouira (2005) and Cavalcanti et al. (2007). In addition, we implement a precise and efficient code to study the behavior of the transmission problem when k 1 ≠ k 2 and when one has a nonlinear frictional dissipation g ( u t ) . More precisely, we aim to numerically check the general decay rate estimates of the energy associated to the problem established in first part of the paper.