We describe and test an easy-to-implement two-step high-order compact (2SHOC) scheme for the Laplacian operator and its implementation into an explicit finite-difference scheme for simulating the nonlinear Schrödinger equation (NLSE). Our method relies on a compact ‘double-differencing’ which is shown to be computationally equivalent to standard fourth-order non-compact schemes. Through numeric...

This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier–Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic–parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference...

A fourth-order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two-dimensional Poisson equation in a rectangular domain. Multigrid methods using a partial semicoarsening strategy and line Gauss–Seidel relaxation are designed to solve the resulting sparse linear systems. Numerical experiments are conducted to test the accuracy of t...

We present a fourth-order compact finite difference scheme on the face centered cubic (FCC) grids for the numerical solution of the two-dimensional convection diffusion equation. The seven-point formula is defined on a regular hexagon, where the strategy of directional derivative is employed to make the derivation procedure straightforward, efficient, and concise. A corresponding multigrid meth...

Higher-order compact difference schemes can achieve higher order accuracy on uniform grids. However, in some cases these may not achieve the desired accuracy. Therefore, this paper proposes a multigrid method based on higher-order compact difference scheme for 3D Helmholtz equation on nonuniform grids. Interpolation and restriction operators are designed accordingly. The suggested scheme has up...