Abstract
In this paper, we first present the expression of a model of a fourth-order compact finite difference (CFD) scheme for the convection diffusion equation with variable convection coefficient. Then, we also obtain the fourth-order CFD schemes of the diffusion equation with variable diffusion coefficients. In addition, a fine description of the sixth-order CFD schemes is also developed for equations with constant coefficients, which is used to discuss certain partial differential equations (PDEs) with arbitrary dimensions. In this paper, various ways of numerical test calculations are prepared to evaluate performance of the fourth-order CFD and sixth-order CFD schemes, respectively, and the empirical results are proved to verify the effectiveness of the schemes in this paper.
Highlights
The standard strategy associated with generating higher order finite difference schemes to expand the stencil is proposed by Leonard [1]
In consideration of the problems caused by noncompact finite difference methods, it is desirable to develop a class of schemes involving compact with high order
A great deal of efforts has been devoted to developing the designing schemes of compact finite difference (CFD) for solving various partial differential equations
Summary
The standard strategy associated with generating higher order finite difference schemes to expand the stencil is proposed by Leonard [1]. A new design of sixth-order compact finite-difference method with a nine-point stencil is developed by Nabavi et al [9] to solve the Helmholtz equation in two-dimensional domain under the circumstance of Dirichlet and Neumann boundary. With the idea of the immersed interface method, third- and fourth-order compact finite difference schemes were proposed for solving the Helmholtz equations with discontinuous coefficient [10, 11]. It is worth noting that the advert of [12, 13] a new high-order finite difference discretization strategy, based on the Richardson extrapolation technique and an operator interpolation scheme, is explored to solve convection diffusion equations [14] which exploits an innovative adaptive scheme in terms of Adaptive Mesh Refinement (AMR) and Multigrid Algorithms to achieve a settlement of the fourthorder two-dimensional Poisson equation. A great number of studies reported by Bilbao and Hamilton [15] provides a two-step schemes (which operate over three time levels) of higher order accurate finite difference schemes applied for the wave equation in any number of spatial dimensions
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