Abstract
In this paper, we develop a suitable multigrid iterative solution method for the numerical solution of second- and third-order discrete schemes for the tempered fractional diffusion equation. Our discretizations will be based on tempered weighted and shifted Grünwald difference (tempered-WSGD) operators in space and the Crank–Nicolson scheme in time. We will prove, and show numerically, that a classical multigrid method, based on direct coarse grid discretization and weighted Jacobi relaxation, performs highly satisfactory for this type of equation. We also employ the multigrid method to solve the second- and third-order discrete schemes for the tempered fractional Black–Scholes equation. Some numerical experiments are carried out to confirm accuracy and effectiveness of the proposed method.
Highlights
Multigrid methods have been applied to solving fractional diffusion equations (FDE) in the literature [9,10,11,12,13]
Hamid et al constructed multigrid methods for a two-dimensional FDE problem, which was discretized by means of a CN-weighted and shifted Grünwald type difference (WSGD) scheme, and they confirmed that multigrid methods performed better than classical preconditioners based on multilevel circulant matrices, in [13]
We provide a multigrid method to solve the presented linear systems originating from the discretized fractional diffusion equations
Summary
Fractional Diffusion Equations.In this paper, we will develop a multigrid method to numerically solve, highly efficiently, the tempered fractional diffusion equation. Fractional diffusion equations are governed by their long range interactions, so that, after discretization, full matrices result. These full matrices may possess a favorable structure, like a Toeplitz matrix structure, which is beneficial regarding efficient matrix-vector multiplication. Hamid et al constructed multigrid methods for a two-dimensional FDE problem, which was discretized by means of a CN-WSGD scheme, and they confirmed that multigrid methods performed better than classical preconditioners based on multilevel circulant matrices, in [13]. Et al [14] reformulated the classical time-stepping schemes as a kind of parallel-in-time (PinT) methods for both one- and two-dimensional space fractional diffusion equations and the fast Krylov subspace method with tau preconditioners is used to solve the resulting discretized linear systems
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