Abstract
The combined compact difference (CCD) scheme has better spectral resolution than many other existing compact or noncompact high-order schemes, and is widely used to solve many differential equations. However, due to its implicit nature, very little theoretical results on the CCD method are known. In this paper, we provide a rigorous theoretical proof for the unique solvability of the CCD scheme for solving the convection-diffusion equation with variable convection coefficients subject to periodic boundary conditions.
Highlights
In many real physical applications, performing high-order and efficient numerical methods for solving the partial differential equations is essential
The compact difference scheme (CCD) scheme, which can be regarded as an extension of the standard Pade schemes as discussed by Lele [1], allows us to conveniently handle the differential equations with variable coefficients subject to Robin boundary conditions
Zhang [32] derived the truncation error representation of the CCD scheme when applied to 1D convection-diffusion equations and analyzed its oscillation property
Summary
1 Introduction In many real physical applications, performing high-order and efficient numerical methods for solving the partial differential equations is essential. When using the CCD method to solve the differential equations, the equation is assumed to be valid at the boundary, and the first and second derivatives together with the function values of unknowns at grid points are computed simultaneously [2]. The CCD method is originally proposed to solve second-order linear ordinary differential equations [2]. Wang et al Advances in Difference Equations (2018) 2018:163 ing direction implicit (ADI) technique to convert it into a series of one-dimensional (1D) problems, which can be solved efficiently by the CCD scheme [16,17,18,19,20,21,22].
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