Error estimates for semi-Lagrangian finite difference methods applied to Burgers' equation in one dimension

Abstract

We make an analytic study of the diffusive, dispersive and overall errors, which arise when using semi-implicit semi-Lagrangian (SISL) finite difference methods to approximate those travelling wave solutions of the one-dimensional Burgers' equation with small diffusion, which develop sharp fronts. For the case of a fixed uniform spatial mesh, with piecewise linear interpolation, a backward error analysis approach is used to construct a precise formal analytic description of the front profile of the numerical approximation to this solution. From this description it is possible to obtain precise estimates of the front width and the front speed in terms of the spatial and temporal step size and to express the overall solution error in terms of these. These formal estimates agree closely with numerical calculations, both qualitatively and quantitatively, and display a roughly periodic behaviour as the number Nx of mesh points increases, and the CFL number passes through integer values. In particular, they show that despite the otherwise poor resolution of the method, the front width is closely approximated when the CFL number is close to an integer, and the front speed is closely approximated when it is close to a half integer. The overall L2 error also shows super-convergence for certain values of Nx. This possibly motivates doing two calculations with different Nx when using the SISL method on such problems to separately minimise the diffusive and dispersive errors. Similar errors in the front width and speed are observed for a number of different interpolation schemes with and without flux limiters.