An efficient implementation of fourth-order compact finite difference scheme for Poisson equation with Dirichlet boundary conditions. (English) Zbl 1443.65288
Summary: Fourth-order compact finite difference scheme has been proposed for solving the Poisson equation with Dirichlet boundary conditions for some time. An efficient implementation of such numerical scheme is often desired for practical usage. In this paper, based on fast discrete Sine transform, we design an efficient algorithm to implement this scheme. To do this, Poisson equation is first discretized by fourth-order compact finite difference method. The subsequent discretized system is not solved by the usual method-matrix inversion, instead it is solved with the fast discrete Sine transform. By doing this way, the computational cost of proposed algorithm for such scheme with large grid numbers can be greatly reduced. Detailed numerical algorithm of this fast solver for one-dimensional, two-dimensional and three dimensional Poisson equation has been presented. Numerical results in one dimension, two dimensions, three dimensions and four dimensions have shown that the applied compact finite difference scheme has fourth order accuracy and can be efficiently implemented.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
| 35J25 | Boundary value problems for second-order elliptic equations |
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