Application of High-Order Compact Difference Schemes for Solving Partial Differential Equations with High-Order Derivatives

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 L-2 norm error, which in some cases reaches a machine precision already on relatively coarse computational meshes.