Abstract: Publication date: Available online 13 March 2020Source: Applied Numerical MathematicsAuthor(s): M. Esmaeilbeigi, O. Chatrabgoun, A. Hosseinian-Far, R. Montasari, A. Daneshkhah

Abstract: Publication date: Available online 12 March 2020Source: Applied Numerical MathematicsAuthor(s): A. Karimi Dizicheh, S. Salahshour, A. Ahmadian, D. Baleanu

Abstract: Publication date: Available online 6 March 2020Source: Applied Numerical MathematicsAuthor(s): Mingfa Fei, Nan Wang, Chengming Huang, Xiaohua Ma

Abstract: Publication date: Available online 28 February 2020Source: Applied Numerical MathematicsAuthor(s): Jamilu Sabi'u, Abdullah Shah, Mohammed Yusuf Waziri

Abstract: Publication date: Available online 26 February 2020Source: Applied Numerical MathematicsAuthor(s): H. Temimi, M. Ben-Romdhane, M. Baccouch, M.O. Musa

Abstract: Publication date: July 2020Source: Applied Numerical Mathematics, Volume 153Author(s): M.H. Heydari This paper develops an accurate computational method based on the shifted Chebyshev cardinal functions (CCFs) for a new class of systems of nonlinear variable-order fractional quadratic integral equations (QIEs). In this way, a new operational matrix (OM) of variable-order fractional integration is obtained for these cardinal functions. In the proposed method, the unknown functions of a system of nonlinear variable-order fractional QIEs are approximated by the shifted CCFs with undetermined coefficients. Then, these approximations are substituted into the system. Next, the OM of variable-order fractional integration and the cardinal property of the shifted CCFs are utilized to reduce the system into an equivalent system of nonlinear algebraic equations. Finally, by solving this algebraic system an approximate solution for the problem is obtained. The main idea behind this approach is to reduce such problems to solving systems of nonlinear algebraic equations, which greatly simplifies the problem. Convergence of the presented method is investigated theoretically and numerically. Furthermore, the proposed approach is numerically evaluated by solving some test problems.

Abstract: Publication date: Available online 21 February 2020Source: Applied Numerical MathematicsAuthor(s): Vahid Mohammadi, Mehdi Dehghan In the current research paper, the generalized moving least squares technique is considered to approximate the spatial variables of two time-dependent phase field partial differential equations on the spheres in Cartesian coordinate. This is known as a direct approximation (it is the standard technique for generalized finite difference scheme [69], [78]), and it can be applied for scattered points on each local sub-domain. The main advantage of this approach is to approximate the Laplace-Beltrami operator on the spheres using different types of distribution points simply, in which the studied mathematical models are involved. In fact, this scheme permits us to solve a given partial differential equation on the sphere directly without changing the original problem to a problem on a narrow band domain with pseudo–Neumann boundary conditions. A second-order semi-implicit backward differential formula (by adding a stabilized term to the chemical potential that is the second-order Douglas-Dupont-type regularization) is applied to approximate the temporal variable. We show that the time discretization considered here guarantees the mass conservation and energy stability. Besides, the convergence analysis of the proposed time discretization is given. The resulting full-discrete scheme of each partial differential equation is a linear system of algebraic equations per time step that is solved via an iterative method, namely biconjugate gradient stabilized algorithm. Some numerical experiments are presented to simulate the phase field Cahn-Hilliard, nonlocal Cahn-Hilliard (for diblock copolymers as microphase separation patterns) and crystal equations on the two-dimensional spheres.

Abstract: Publication date: Available online 21 February 2020Source: Applied Numerical MathematicsAuthor(s): Anthony C. Brooms, Theodore C. Holtom We consider the problem of quantifying uncertainty for converging beam triple LIDAR when the input uncertainty follows a uniform distribution. We determine expressions for the range (i.e. set of reachable points) for the reconstructed velocity vector as a function of any particular setting of the nominal input parameters and determine an explicit lower (and upper) bound on the (averaged) volume (with respect to Lebesgue measure), in R3, of that range. We show that the size of any such bound is inversely proportional to the absolute value of the triple scalar product of the unit vectors characterizing the Doppler measurement directions (optimized over the uncertainty region) in R6 associated with the nominal angle settings under consideration. This leads to the conclusion that the nominal LIDAR configurations that minimize output uncertainty ought to be those in which the value of the triple scalar product of the Doppler unit vectors is at its largest.

Abstract: Publication date: Available online 21 October 2019Source: Applied Numerical MathematicsAuthor(s): M.N. Kapetina, A. Pisano, M.R. Rapaić, E. Usai A continuation of previous authors' work on adaptive parameter estimation for linear dynamical systems having irrational transfer function is presented in this work. An original modification of the gradient algorithm, inspired by the variable structure control techniques and additionally featuring a time-varying adaptation gain, is presented and analyzed using Lyapunov techniques. The exposition is illustrated by several numerical examples which illustrate the effectiveness of the proposed algorithm.

Abstract: Publication date: Available online 19 September 2019Source: Applied Numerical MathematicsAuthor(s): Sabrina Francesca Pellegrino In this paper we validate the implementation of the numerical scheme proposed in [3]. The validation is made by comparison with an explicit solution here obtained, and the solutions of Riemann problems for several networks. We then perform some simulations in order to qualitatively validate the model under consideration.Such results represent also a first step for the validation of the finite volumes scheme introduced in [14].

Abstract: Publication date: Available online 24 July 2019Source: Applied Numerical MathematicsAuthor(s): M. Abundo, G. Ascione, M.F. Carfora, E. Pirozzi We show that the First-Passage-Time probability distribution of a Lévy time-changed Brownian motion with drift is solution of a time fractional advection-diffusion equation subject to initial and boundary conditions; the Caputo fractional derivative with respect to time is considered. We propose a high order compact implicit discretization scheme for solving this fractional PDE problem and we show that it preserves the structural properties (non-negativity, boundedness, time monotonicity) of the theoretical solution, having to be a probability distribution. Numerical experiments confirming such findings are reported. Simulations of the sample paths of the considered process are also performed and used to both provide suitable boundary conditions and to validate the numerical results.

Abstract: Publication date: Available online 12 July 2019Source: Applied Numerical MathematicsAuthor(s): Luca Dieci In this work we show that numerical integration during sliding motion, for piecewise-smooth systems, can be performed without the need to project the approximate solutions on the constraints' surface. We give a general algorithm, an order of approximation result, and show the effectiveness of the technique on several examples. Comparison between projected and non-projected methods and the standard regularization approach is also given.

Abstract: Publication date: Available online 18 June 2019Source: Applied Numerical MathematicsAuthor(s): F. Iavernaro, F. Mazzia, M.S. Mukhametzhanov, Ya.D. Sergeyev Multi-derivative one-step methods based upon Euler–Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge–Kutta methods, we show that the Euler–MacLaurin method of order p is conjugate-symplectic up to order p+2. This feature entitles them to play a role in the context of geometric integration and, to make their implementation competitive with the existing integrators, we explore the possibility of computing the underlying higher order derivatives with the aid of the Infinity Computer.

Abstract: Publication date: Available online 23 May 2019Source: Applied Numerical MathematicsAuthor(s): Eleonora Messina, Eugenio Chioccarelli, Georgios Baltzopoulos, Antonia Vecchio The dynamic behaviour of rigid blocks subjected to support excitation is represented by discontinuous differential equations with state jumps. In the numerical simulation of these systems, the jump times corresponding to the numerical trajectory do not coincide with the ones of the given problem. When multiple state jumps occur, this approximation may affect the accuracy of the solution and even cause an order reduction in the method. Focus here is on the error behaviour in the numerical dynamic. The basic idea is to investigate how the error propagates in successive impacts by decomposing the numerical integration process of the overall system into a sequence of discretized perturbed problems.

Abstract: Publication date: Available online 12 April 2019Source: Applied Numerical MathematicsAuthor(s): Lorenzo Mascotto, Alexander Pichler We extend the nonconforming Trefftz virtual element method introduced in [30] to the case of the fluid-fluid interface problem, that is, a Helmholtz problem with piecewise constant wave number. With respect to the original approach, we address two additional issues: firstly, we define the coupling of local approximation spaces with piecewise constant wave numbers; secondly, we enrich such local spaces with special functions capturing the physical behavior of the solution to the target problem. As these two issues are directly related to an increase of the number of degrees of freedom, we use a reduction strategy inspired by [31], which allows to mitigate the growth of the dimension of the approximation space when considering h- and p-refinements. This renders the new method highly competitive in comparison to other Trefftz and quasi-Trefftz technologies tailored for the Helmholtz problem with piecewise constant wave number. A wide range of numerical experiments, including the p-version with quasi-uniform meshes and the hp-version with isotropic and anisotropic mesh refinements, is presented.

Abstract: Publication date: Available online 3 April 2019Source: Applied Numerical MathematicsAuthor(s): F. Dassi, G. Vacca We present the essential tools to deal with virtual element method (VEM) for the approximation of solutions of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep analysis on polynomial spaces and decompositions. We exploit such “bricks” to construct virtual element approximations of Stokes, Darcy and Navier–Stokes problems and we provide a series of examples to numerically verify the theoretical behaviour of high-order VEM.

Abstract: Publication date: July 2020Source: Applied Numerical Mathematics, Volume 153Author(s): Wei Liu, Xuerong Mao, Jingwen Tang, Yue Wu The truncated Euler-Maruyama (EM) method is proposed to approximate a class of non-autonomous stochastic differential equations (SDEs) with the Hölder continuity in the temporal variable and the super-linear growth in the state variable. The strong convergence with the convergence rate is proved. Moreover, the strong convergence of the truncated EM method for a class of highly non-linear time-changed SDEs is studied.

Abstract: Publication date: Available online 14 February 2020Source: Applied Numerical MathematicsAuthor(s): Sio Wan Ng, Siu-Long Lei, Juan Lu, Zhiguo Gong SimRank is popularly used for evaluating similarities between nodes in a graph. Though some recent work has transformed the SimRank into a linear system which can be solved by using conjugate gradient (CG) algorithm. However, the diagonal preconditioner used in the algorithm still left some room for improvement. There is no work showing what should be an optimal preconditioner for the linear system. For such a sake, in this paper, we theoretically study the preconditioner problem and propose a polynomial preconditioner which can improve the computation speed significantly. In our investigation, we also modify the original linear system to adapt to all situations of the graph. We further prove that the condition number of a polynomial preconditioned matrix is bounded in a narrow interval. It implies that the new preconditioner can effectively accelerate the convergence rate of the CG algorithm. The experimental results show the proposed algorithm outperforms the state-of-the-art algorithms for the all-pairs SimRank computation.

Abstract: Publication date: Available online 13 February 2020Source: Applied Numerical MathematicsAuthor(s): Wansheng Wang, Jiao Tang The design of efficient numerical methods, which produce an accurate approximation of the solutions, for solving time-dependent Schrödinger equation in the semiclassical regime, where the Planck constant ε is small, is a formidable mathematical challenge. In this paper a new method is shown to construct exponential splitting schemes for linear time-dependent Schrödinger equation with a linear potential. The local discretization error of the two time-splitting methods constructed here is O(max{Δt3,Δt5/ε}), while the well-known Lie-Trotter splitting scheme and the Strang splitting scheme are O(Δt2/ε) and O(Δt3/ε), respectively, where Δt is the time step-size. The global error estimates of new exponential splitting schemes with spectral discretization suggests that larger time step-size is admissible for obtaining high accuracy approximation of the solutions. Numerical studies verify our theoretical results and reveal that the new methods are especially efficient for linear semiclassical Schrödinger equation with a quadratic potential.

Abstract: Publication date: Available online 13 February 2020Source: Applied Numerical MathematicsAuthor(s): Hengguang Li, Qinghui Zhang Meshfree methods (MMs) enjoy advantages in discretizing problem domains over mesh-based methods. Extensive progress has been made in the development of the MMs in the last three decades. The commonly used MMs, such as the reproducing kernel particle methods (RKP), the moving least-square methods (MLS), and the meshless Petrov-Galerkin methods, have main difficulties in numerical integration and in imposing essential boundary conditions (EBC). Motivated by conventional finite volume methods, we propose a meshfree finite volume method (MFVM), where the trial functions are constructed through the conventional RKP or MLS procedures, while the test functions are set to be piesewise constants on Voronoi diagrams built on scattered particles. The proposed method possesses three typical merits: (1) the standard Gaussian rules are proven to produce optimal approximation errors; (2) the EBC can be imposed directly on boundary particles; and (3) mass conservation is maintained locally due to its finite volume formulations. Inf-sup conditions for the MFVM are proven in a one-dimensional problem, and are demonstrated numerically using a generalized eigenvalue problem for higher dimensions. Numerical test results are reported to verify the theoretical findings.

Abstract: Publication date: Available online 13 February 2020Source: Applied Numerical MathematicsAuthor(s): Lynn Boen, Karel J. in 't Hout This paper deals with the efficient numerical solution of the two-dimensional partial integro-differential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Merton jump-diffusion model. We consider the adaptation of various operator splitting schemes of both the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind that have recently been studied for partial integro-differential equations (PIDEs) in [3]. Each of these schemes conveniently treats the nonlocal integral part in an explicit manner. Their adaptation to PIDCPs is achieved through a combination with the Ikonen–Toivanen splitting technique [14] as well as with the penalty method [32]. The convergence behaviour and relative performance of the acquired eight operator splitting methods is investigated in extensive numerical experiments for American put-on-the-min and put-on-the-average options.

Abstract: Publication date: Available online 12 February 2020Source: Applied Numerical MathematicsAuthor(s): V. Dhanya Varma, Nagaiah Chamakuri, Suresh Kumar Nadupuri In this work, the Discontinuous Galerkin (DG) schemes were studied for the solution of convection-diffusion-reaction equations which arise from mathematical modeling of fluidized bed spray granulation process (FBSG). The discontinuous Galerkin method for space discretization is employed to treat the dominated convection behavior in the governing equations.The implicit Euler method for the temporal discretization is used. The mathematical analysis of the governing equations with a priori bounds for all the state variables is demonstrated. The investigation of experimental order of convergence is presented for test functions of degrees one and two. Finally, parallel efficiency of strong scaling is presented for the employed DG schemes.

Abstract: Publication date: July 2020Source: Applied Numerical Mathematics, Volume 153Author(s): Hui Duan, Xinjuan Chen, Jae-Hun Jung Fractional differential equations have become an important modeling technique in describing various natural phenomena. A variety of numerical methods for solving fractional differential equations has been developed over the last decades. Among them, finite difference methods are most popular owing to relative easiness for implementation.In this paper, we show that the finite difference method with the Riemann-Liouville (RL) fractional derivative yields inconsistent and oscillatory numerical solutions to fractional differential equations of discontinuous problems for the fractional order α, 0

Abstract: Publication date: Available online 11 February 2020Source: Applied Numerical MathematicsAuthor(s): Ankita Shukla, Mani Mehra In this paper, the backward heat conduction problem is solved numerically using a fourth-order compact difference scheme. For the regularization of this ill-posed problem, fourth-order compact filtering which is a spatial filtering technique is proposed. The second derivative approximation in compact difference scheme has been compared with the fourth-order standard finite difference scheme via Fourier analysis. The comparison shows that the resolution ability of the compact difference scheme is better than the standard finite difference scheme. Meanwhile, an error estimate between the exact and regularized solution is derived. Numerical results are provided using the proposed method. The comparison of CPU time shows that the fourth-order compact difference scheme takes lesser CPU time than the fourth-order standard finite difference scheme.

Abstract: Publication date: Available online 10 February 2020Source: Applied Numerical MathematicsAuthor(s): Yonghui Qin, Heping Ma A Legendre-tau-Galerkin method is developed for nonlinear evolution problems and its multiple interval form is also considered. The Legendre tau method is applied in time and the Legendre/Chebyshev–Gauss–Lobatto points are adopted to deal with the nonlinear term. By taking appropriate basis functions, it leads to a simple discrete equations. The proposed method enables us to derive optimal error estimates in L2-norm for the Legendre collocation under the two kinds of Lipschitz conditions, respectively. Our method is also applied to the numerical solutions of some nonlinear partial differential equations by using the Legendre Galerkin and Chebyshev collocation in spatial discretization. Numerical examples are given to show the efficiency of the methods.

Abstract: Publication date: Available online 6 February 2020Source: Applied Numerical MathematicsAuthor(s): Zhong Du, Bo Tian, Qi-Xing Qu, Xiao-Yu Wu, Xue-Hui Zhao Non-Kerr media possess certain applications in photonic lattices and optical fibers. Studied in this paper are the vector rational and semi-rational rogue waves in a non-Kerr medium, through the coupled cubic-quintic nonlinear Schrödinger system, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium. Applying the gauge transformations, we derive the Nth-order Darboux transformtion and Nth-order vector rational and semi-rational rogue wave solutions, where N is a positive integer. With such solutions, we present three types of the second-order rogue waves with the triangle structure: the one with each component containing three four-petalled rogue waves, the one with each component containing three eye-shaped rogue waves, and the other with one component containing three anti-eye-shaped rogue waves and the other component containing three eye-shaped rogue waves. We exhibit the third-order vector rogue waves with the merged, triangle and pentagon structures in each component. Moreover, we show the first- and second-order vector semi-rational rogue waves which display the coexistence of the rogue waves and the breathers.

Abstract: Publication date: Available online 4 February 2020Source: Applied Numerical MathematicsAuthor(s): Philipp Öffner, Davide Torlo Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Finally, we validate our theoretical analysis through numerical simulations.

Abstract: Publication date: Available online 4 February 2020Source: Applied Numerical MathematicsAuthor(s): Zhihao Ge, Yanan He, Yinnian He In the paper, a new lowest equal-order stabilized mixed finite element method is proposed for a poroelasticity model in displacement-pressure formulation, which is based on multiphysics approach. The original model is reformulated to reveal the multi-physical process of deformation and diffusion and get a coupled fluid system. Then, a time-stepping algorithm which decouples the reformulated problem at each time step and the lowest equal-order stabilized mixed finite element method for the reformulated problem is given, which can overcome the “locking” phenomenon. Also, the stability analysis and error analysis are proved that the stabilized mixed finite element method is stable for the pair of finite elements without the inf-sup condition and has the optimal convergence order. Finally, the numerical examples are shown to verify the theoretical results, and a conclusion is drawn to summarize the main results in this paper.

Abstract: Publication date: Available online 24 January 2020Source: Applied Numerical MathematicsAuthor(s): Fang Chen, Tian-Yi Li, Kang-Ya Lu, Galina V. Muratova We introduce a modified QHSS (MQHSS) iteration method for solving a class of complex symmetric linear systems, and prove that this MQHSS iteration method is convergent under certain conditions. We also consider acceleration of the MQHSS iteration by Krylov subspace methods. Numerical experiments show the good performance of the MQHSS iteration method.

Abstract: Publication date: Available online 23 January 2020Source: Applied Numerical MathematicsAuthor(s): Alessandra Jannelli, Marianna Ruggieri, Maria Paola Speciale In this paper, numerical solutions of space-fractional advection-diffusion equations, involving the Riemann-Liouville derivative with a nonlinear source term, are presented. We propose a procedure that combines the fractional Lie symmetries analysis, to reduce the original fractional partial differential equations into fractional ordinary differential equations, with a numerical method. By adopting the Caputo definition of derivative, the reduced fractional ordinary equations are solved by applying the implicit trapezoidal method. The numerical results confirm the applicability and the efficiency of the proposed approach.

Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): M. Falcone, S. Finzi VitaAbstractWe present a new two-layer closed form model for the dynamics of desert dunes under the effect of a horizontal wind blowing in an arbitrary direction. This model is an extension of a very simplified model previously introduced by Hadeler and Kuttler [12]. Our extension, inspired by the sandpile dynamics approach, includes the effects of gravity on both sides (upwind and downwind) of the dune, and allows to describe erosion and deposition in a more accurate way. After a discussion of the model and its properties we present a numerical scheme based on finite differences in 1D and we prove its consistency and stability. Some numerical tests show a good qualitative behavior and a realistic shape for the evolving dunes. Finally, we discuss the preliminary steps of a possible extension of this model to the 2D case.

Abstract: Publication date: Available online 19 December 2019Source: Applied Numerical MathematicsAuthor(s): Angela Martiradonna, Gianpiero Colonna, Fasma DieleAbstractWe generalize the nonstandard Euler and Heun schemes in order to provide explicit geometric numerical integrators for biochemical systems, here denoted as GeCo schemes, that preserve both positivity of the solutions and linear invariants. We relax the request on the order convergence of the denominator function for the first-order approximation and we let it depend on the step size also throughout the solution approximating values. The first-order variant is exact on a two-dimensional linear test problem. Moreover, we introduce a class of modified mGeCo(α) schemes and, by tuning the parameter α≥1, we improve the numerical performance of GeCo integrators on some examples taken from the literature. A numerical comparison with BBKS and mBBKS schemes, which are implicit integrators, positive and conserve linear invariants, show the gain in efficiency of both GeCo and mGeCo(α) procedures as they generate similar errors with an explicit functional form.

Abstract: Publication date: Available online 5 December 2019Source: Applied Numerical MathematicsAuthor(s): Alessandro Alla, Luca SaluzziAbstractThe Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation.Here, we aim at testing the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests.