Abstract
In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.
Highlights
Various powerful mathematical methods such as the homotopy perturbation method, variational iteration method, Adomian decomposition method and others [1,2,3] have been proposed to obtain approximate solutions in partial differential equations (PDEs)
We briefly describe preconditioners that we use for solving linear systems and matrix A is block tri-diagonal
We study comparison of different preconditioners in combination Krylov subspace methods
Summary
Various powerful mathematical methods such as the homotopy perturbation method, variational iteration method, Adomian decomposition method and others [1,2,3] have been proposed to obtain approximate solutions in partial differential equations (PDEs). We accomplish a comprehensive study for different preconditioners in combination with Krylov subspace methods for solving linear systems arising from the compact finite difference schemes [10, 11] for 2-D parabolic equation uxx uyy f (x, y,t,u,ux , uy , ut ) (1.1). Krylov subspace methods are one of the widely used and successful classes of numerical algorithms for solving large and sparse systems of algebraic equations but the speed of these methods are slow for problems which arise from typical applications [12,13,14,15]. We accomplish a comprehensive study for different preconditioners in combination with Krylov subspace methods for solving linear systems arising from the compact high-order approximations. We present the results of our comparative study in the final section
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