High order WENO schemes: investigations on non-uniform convergence for MHD Riemann problems

Detailed empirical error and convergence studies for high order weighted essentially non-oscillatory (WENO) schemes are presented in the case of special magnetohydrodynamic Riemann problems. The results supplement the results for standard high resolution finite volume schemes given in [M. Torrilhon, Non-uniform convergence of finite-volume-schemes for Riemann problems of ideal magnetohydrodynamics, J. Comput. Phys. 192 (2003) 73–94]. The special Riemann problems are based on initial conditions that are close to initial conditions which admit non-unique solutions. Like the standard methods the WENO schemes investigated in this paper exhibit a strongly non-uniform convergence behavior with initial convergence to a wrong solution (pseudo-convergence). However, the cancelation of the pseudo-convergence occurs at coarser grids for higher order methods.