Spatial convergence study of Lax-Friedrichs WENO fast sweeping method on the SN transport equation with nonsmoothness

Abstract

Spatial convergence study of the Lax-Friedrichs fast sweeping method based on the classical third order WENO scheme (WENO3) without a priori smoothness indicators is performed on the variants of Larsen’s benchmark problem with nonsmooth solutions. For comparison, third order upwind (UPWD3), weighted diamond difference, step method, bilinear discontinuous finite element method are also included. In optically thick regimes of the cases with continuous angular fluxes, WENO3 detects steep gradients of angular fluxes and handles them as discontinuities to suppress the unphysical oscillation, which improves the accuracy significantly compared with UPWD3. In the cases with discontinuous angular fluxes, WENO3 achieves non-oscillatory solutions with limited numerical diffusion. Nonlinear Gauss-Seidel iteration is necessary to obtain convergent solution for the nonlinearity of WENO3, which leads to low computational efficiency. The convergence behavior of nonlinear Gauss-Seidel iteration is affected by nonsmoothness of the exact solution, mesh number and the relaxation parameter.