High-Order Compact Difference Method On Non-Uniform Grids For Solving The Linear Hyperbolic Equations

High-order compact difference method attracts more and more researchers’ interests for its higher order accuracy, less grid points needed and easier treatment of boundary conditions. A great deal of research work has been published to the numerical simulations in computational fluid dynamics and the numerical solutions to partial differential equations. However, we notice that the most high order compact difference schemes are built on uniform grids, and mainly for elliptic equations. For non-elliptic equations, there are few high order compact difference schemes are developed on non-uniform grids.Based on the proposed high-order compact difference schemes on uniform grids for the hyperbolic equations, high-order compact schemes on the non-uniform grids are developed in this paper. At first, the high-order compact scheme on the non-uniform grids for the1D linear hyperbolic equation is proposed. The scheme is the second order accuracy in time and the third to fourth order accuracy in space. Then, based on the proposed high-order accurate compact alternating direction implicit (ADI) difference schemes on uniform grids, a high-order compact ADI scheme on the non-uniform grids for the2D linear hyperbolic equation is given. The scheme is the second order accuracy in time and the third to fourth order accuracy in space. At last, the high-order compact ADI method is generalized to the3D linear hyperbolic equation. Numerical experiments are conduct to validate the high-order compact schemes proposed in this paper and numerical results show that the high order compact schemes on non-uniform grids are superior than those on uniform grids when the problems with great gradients and boundary layers are considered.