A stable Gaussian radial basis function method for solving nonlinear unsteady convection-diffusion-reaction equations

被引:32
作者
Rashidinia, J. [1 ]
Khasi, M. [1 ]
Fasshauer, G. E. [2 ]
机构
[1] Iran Univ Sci & Technol, Sch Math, Tehran, Iran
[2] Colorado Sch Mines, Dept Appl Math & Stat, Golden, CO 80401 USA
基金
美国国家科学基金会;
关键词
Radial basis functions; Eigenfunction expansion; Matrix system of ODEs; Convection-diffusion-reaction equations; Robin boundary condition; Adams-Bashforth; FINITE-ELEMENT METHODS; MATRICES; COMPUTATION; ALGORITHM;
D O I
10.1016/j.camwa.2017.12.007
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We investigate a novel method for the numerical solution of two-dimensional time-dependent convectiondiffusionreaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor-corrector schemes such as Adams-Bashforth and implicit-explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems. (C) 2017 Elsevier Ltd. All rights reserved.
引用
收藏
页码:1831 / 1850
页数:20
相关论文
共 32 条
[1]  
[Anonymous], 2000, SIAM
[2]  
[Anonymous], 2008, Functions of matrices: theory and computation
[3]  
[Anonymous], 2012, MATRIX COMPUTATIONS
[4]   IMPLICIT EXPLICIT METHODS FOR TIME-DEPENDENT PARTIAL-DIFFERENTIAL EQUATIONS [J].
ASCHER, UM ;
RUUTH, SJ ;
WETTON, BTR .
SIAM JOURNAL ON NUMERICAL ANALYSIS, 1995, 32 (03) :797-823
[5]   ALGORITHM - SOLUTION OF MATRIX EQUATION AX+XB = C [J].
BARTELS, RH ;
STEWART, GW .
COMMUNICATIONS OF THE ACM, 1972, 15 (09) :820-&
[6]   Jacobi-Gauss-Lobatto collocation method for solving nonlinear reaction-diffusion equations subject to Dirichlet boundary conditions [J].
Bhrawy, A. H. ;
Doha, E. H. ;
Abdelkawy, M. A. ;
Van Gorder, R. A. .
APPLIED MATHEMATICAL MODELLING, 2016, 40 (03) :1703-1716
[7]   An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels [J].
Cavoretto, Roberto ;
Fasshauer, Gregory E. ;
McCourt, Michael .
NUMERICAL ALGORITHMS, 2015, 68 (02) :393-422
[8]   The solution of two-dimensional advection-diffusion equations via operational matrices [J].
de la Hoz, Francisco ;
Vadillo, Fernando .
APPLIED NUMERICAL MATHEMATICS, 2013, 72 :172-187
[9]   Accurate SVDs of polynomial Vandermonde matrices involving orthonormal polynomials [J].
Demmel, James ;
Koev, Plamen .
LINEAR ALGEBRA AND ITS APPLICATIONS, 2006, 417 (2-3) :382-396
[10]   STABLE EVALUATION OF GAUSSIAN RADIAL BASIS FUNCTION INTERPOLANTS [J].
Fasshauer, Gregory E. ;
Mccourt, Michael J. .
SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2012, 34 (02) :A737-A762