Numerical solution of fractional diffusion-reaction problems based on BURA
| Autor: | Stanislav Harizanov, Raytcho D. Lazarov, Pencho Marinov, Svetozar Margenov |
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| Přispěvatelé: | Pencho Marinov |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Linear system Finite difference 010103 numerical & computational mathematics Positive-definite matrix 01 natural sciences 010101 applied mathematics Computational Mathematics Matrix (mathematics) Computational Theory and Mathematics Error analysis Modeling and Simulation Fractional diffusion-reaction 0101 mathematics Algebraic number Best uniform rational approximation Scalar field Scaling Eigenvalues and eigenvectors Mathematics |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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