On the convergence of compact difference schemes

Abstract

Difference schemes that are compact in space, i.e., schemes constructed on a two- or three-point stencil in each spatial direction, are more efficient and convenient for boundary condition formulation than other high-order accurate schemes. Originally, these schemes were developed primarily to obtain smooth solutions. In the last two decades, compact schemes have been actively used to compute gas dynamic flows with shock waves. However, when a numerical solution with guaranteed accuracy is desired, the actual properties of difference schemes have to be known in the calculation of solutions with discontinuities. For some widely used compact schemes, this issue has not yet been well studied. The properties of compact schemes constructed by the method of lines are examined in this paper. An initial-boundary value problem for the linear heat equation with discontinuous initial data is used as a test problem. In the method of lines, the spatial derivative in the heat equation is approximated on a two-point stencil according to a fourth-order accurate compact differentiation formula. The resulting evolution system of ordinary differential equations is solved using various implicit one-step two- and three-stage schemes of the second and third order of accuracy. The relation between the properties of the stability function of a scheme and the spatial monotonicity of the numerical solution is analyzed. In computations over long time intervals, the compact schemes are shown to be superior to traditional schemes based on the second-order accurate three-point approximation of the spatial derivative.