Numerical solution of fractional diffusion–reaction problems based on BURA. (15th July 2020) - Harizanov, Stanislav; Lazarov, Raytcho; Margenov, Svetozar; Marinov, Pencho
Abstract: The paper is devoted to the numerical solution of algebraic systems of the type ( A α + q I ) u = f, 0 < α < 1, q > 0, u, f ∈ R N, where A is a symmetric and positive definite matrix. We assume that A is obtained by finite difference approximation of a second order diffusion problem in Ω ⊂ R d, d = 1, 2 so that A α + q I approximates the related fractional diffusion–reaction operator or could be a result of a time-stepping procedure in solving time-dependent sub-diffusion problems. We also assume that a method of optimal complexity for solving linear systems with matrices A + c I, c ≥ 0 is available. We analyze and study numerically a class of solution methods based on the best uniform rational approximation (BURA) of a certain scalar function in the unit interval. The first such method, originally proposed in Harizanov et al. (2018) for numerical solution of fractional-in-space diffusion problems, was based on the BURA r α ( ξ ) of ξ 1 − α in [ 0, 1 ] through scaling of the matrix A by its largest eigenvalue. Then the BURA of t − α in [ 1, ∞ ) is given by t − 1 r α ( t ) and correspondingly, A − 1 r α ( A ) is used as an approximation of A − α . Further, this method was improved in Harizanov et al. (2019) using the same concept but by scaling the matrix A by its smallest eigenvalue. In this paper we consider the BURA r α ( ξ ) of 1 ∕ ( ξ − α + q ) for ξ ∈ ( 0, 1 ] . Then we define the approximation of ( A α + q I ) − 1 as r α ( A − α ) . We also propose an alternative method that uses BURA of ξ α to produce certain uniform rational approximation (URA) of 1 ∕ ( ξ − α + q ) . Comprehensive numerical experiments are used to demonstrate the computational efficiency and robustness of the new BURA and URA methods.