Abstract
In this paper, high-order compact-difference schemes involving a large number of mesh points in the computational stencils are used to numerically solve partial differential equations containing high-order derivatives. The test cases include a linear dispersive wave equation, the non-linear Korteweg–de Vries (KdV)-like equations, and the non-linear Kuramoto–Sivashinsky equation with known analytical solutions. It is shown that very high-order compact schemes, e.g., of 20th or 24th orders, cause a very fast drop in the L2 norm error, which in some cases reaches a machine precision already on relatively coarse computational meshes.
Highlights
High-order numerical methods are commonly used in solving mathematical and engineering problems described by the class of higher-order partial differential equations.They are used, among others, in modeling wave propagation, combustion, thermal instability, magnetohydrodynamics, or astrophysics.In computational fluid dynamics, the higher-order derivatives, e.g., of fourth or sixth orders, are used to introduce artificial viscosity terms or to approximate filter functions [1].An accurate approximation of the higher-order derivatives in these problems requires the application of sophisticated and highly accurate methods as, otherwise, the induced errors act as artificially introduced terms destroying the solution accuracy
We focus on compact difference (CD) schemes of very high orders, e.g., 20th or 30th, and demonstrate their accuracy based on canonical test cases
This paper presents a detailed analysis of very high-order compact difference (CD)
Summary
High-order numerical methods are commonly used in solving mathematical and engineering problems described by the class of higher-order partial differential equations. Sivashinsky [43,44,45] in the 1970s for modeling the Belousov–Zhabotinsky reaction and diffusion instability in a laminar flame front It was derived starting from a weakly non-linear and long-wave simplification of the Navier–Stokes equation and so far has been treated as a mathematical model in several different problems, such as the evolution of falling films [46,47], instability between two concurrent viscous flows [48], two-phase flows in cylindrical pipes [49], or drift waves in plasma [50].
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