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Design of High-Order Difference Scheme and Analysis of Solution Characteristics—Part II: A Kind of Third-Order Difference Scheme and New Scheme Design Theory

In this Part II, on the basis of the general style design of second-order difference scheme and the analysis of the absolutely stable scheme proposed in Part I, the companion article, the general design method of any high-order difference scheme is proposed. Based on this method, a new kind of third-order difference scheme including 17 different variants is constructed, which uses the same grid points as existing second-order difference schemes but is different from them in that the grids are chosen symmetrically from two sides of the interface. Because they have the same matrix style created by the same grid plots of the discretization equation, these third-order schemes require the same CPU time and memory as the second-order schemes; however, this kind of symmetrical third-order difference scheme will keep the consistency between the false diffusion and the stability, and the stability of the scheme is better than that of the existing biased second-order scheme. Further research shows that under the conditions of matrix style and computer memory, the scheme constituted by symmetrically numbered grids from two sides of the interface with odd order of accuracy can maintain consistency between numerical accuracy and stability better than any kind of scheme designed according to the “upwind” idea. Based on this understanding, a new scheme design theory called symmetric and odd-order accuracy scheme design theory is proposed.