Special Issue "Numerical Methods for Partial Differential Equations"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 January 2019)

Special Issue Editor

Guest Editor
Prof. Dr. Xinfeng Liu

Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
Website | E-Mail
Interests: Numerical Partial Differential Equations, Computational Systems Biology and Computational Fluid Dynamics

Special Issue Information

Dear Colleagues,

Partial differential equations (PDEs), in general, have attracted more and more attention in mathematical, scientific, and engineering communities due to their wide practical applications in modeling physical/engineering/biological systems of interest in fluid mechanics, aerodynamics, meteorology, combustion, and many other areas. Due to the complexity, it is usually impossible or extremely difficult to solve PDEs analytically. Thus, development of efficient and accurate numerical algorithms for simulation of solutions to these systems continues to be a challenging task. This Special Issue is mainly focused to address a wide range of computational methods ranging from efficient finite element and finite difference methods, adaptive methods, multi-scale methods, to spectral methods and kinetic Monte Carlo simulations. Computational challenges will be discussed, and new computational techniques will be introduced for various applications.

Prof. Dr. Xinfeng Liu
Guest Editor

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Keywords

  • Finite Difference Methods
  • Finite Volume Methods
  • Finite Element Methods
  • Integration Factor Methods
  • Operator-Splitting Methods

Published Papers (7 papers)

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Research

Open AccessArticle
An Efficient Local Formulation for Time–Dependent PDEs
Mathematics 2019, 7(3), 216; https://doi.org/10.3390/math7030216
Received: 30 January 2019 / Revised: 18 February 2019 / Accepted: 20 February 2019 / Published: 26 February 2019
Cited by 2 | PDF Full-text (432 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs. This local approach has flexibility with respect to geometry along with high order of convergence rate. In case [...] Read more.
In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs. This local approach has flexibility with respect to geometry along with high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focused towards localized RBFs approximations, as proposed here. The proposed local meshless procedure is used for spatial discretization, whereas for temporal discretization, different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation on regular and irregular domains. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Open AccessArticle
A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator
Mathematics 2019, 7(2), 113; https://doi.org/10.3390/math7020113
Received: 27 November 2018 / Revised: 6 January 2019 / Accepted: 21 January 2019 / Published: 22 January 2019
Cited by 1 | PDF Full-text (325 KB) | HTML Full-text | XML Full-text
Abstract
An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results obtained for different values of kinematic viscosity are made [...] Read more.
An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results obtained for different values of kinematic viscosity are made with the exact solutions and the reported results to demonstrate the efficiency and accuracy of the algorithm. It is shown that the numerical solutions concur with existing results and the proposed algorithm is efficient and can be easily implemented. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Open AccessArticle
Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations
Mathematics 2018, 6(12), 327; https://doi.org/10.3390/math6120327
Received: 8 October 2018 / Revised: 28 November 2018 / Accepted: 8 December 2018 / Published: 14 December 2018
Cited by 4 | PDF Full-text (1357 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the symmetric radial basis function method is utilized for the numerical solution of two- and three-dimensional elliptic PDEs. Numerical results are obtained by using a set of uniform or random points. Numerical tests are accomplished to demonstrate the efficacy and [...] Read more.
In this paper, the symmetric radial basis function method is utilized for the numerical solution of two- and three-dimensional elliptic PDEs. Numerical results are obtained by using a set of uniform or random points. Numerical tests are accomplished to demonstrate the efficacy and accuracy of the method on both regular and irregular domains. Furthermore, the proposed method is tested for the solution of elliptic PDE in the case of various frequencies. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Open AccessArticle
The Space–Time Kernel-Based Numerical Method for Burgers’ Equations
Mathematics 2018, 6(10), 212; https://doi.org/10.3390/math6100212
Received: 26 September 2018 / Revised: 8 October 2018 / Accepted: 16 October 2018 / Published: 18 October 2018
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Abstract
It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent [...] Read more.
It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent PDEs are collocated in both space and time first, contrary to spatial discretization, and time stepping procedures for time integration are then applied. Physically one cannot in general expect that the spatial and temporal features of the solution behaves on the same order. Hence, one should have to incorporate anisotropic kernels. The nonlinear Burgers’ equations are converted by nonlinear transformation to linear equations. The spatial discretizations are carried out to construct differentiation matrices. Comparisons with most available numerical methods are made to solve the Burgers’ equations. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Open AccessArticle
The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations
Mathematics 2018, 6(10), 211; https://doi.org/10.3390/math6100211
Received: 27 September 2018 / Revised: 11 October 2018 / Accepted: 14 October 2018 / Published: 18 October 2018
Cited by 1 | PDF Full-text (3375 KB) | HTML Full-text | XML Full-text
Abstract
We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can [...] Read more.
We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Open AccessFeature PaperArticle
Numerical Methods for a Two-Species Competition-Diffusion Model with Free Boundaries
Mathematics 2018, 6(5), 72; https://doi.org/10.3390/math6050072
Received: 31 March 2018 / Revised: 27 April 2018 / Accepted: 28 April 2018 / Published: 3 May 2018
Cited by 1 | PDF Full-text (2475 KB) | HTML Full-text | XML Full-text
Abstract
The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of [...] Read more.
The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce a front tracking method coupled with an implicit solver for the 1D model. For the general 2D model, we use a level set approach to handle the moving boundaries to efficiently treat complicated topological changes. Several numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Open AccessArticle
Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations
Mathematics 2017, 5(4), 63; https://doi.org/10.3390/math5040063
Received: 26 September 2017 / Revised: 7 November 2017 / Accepted: 8 November 2017 / Published: 16 November 2017
Cited by 1 | PDF Full-text (565 KB) | HTML Full-text | XML Full-text
Abstract
Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can [...] Read more.
Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can be designed for an arbitrary order of accuracy. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. We analyze the truncation errors of the fully discretized schemes. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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