WENO Schemes for Cylindrical and Spherical Grid

In this paper, we apply the high order WENO schemes to uniform cylindrical and spherical grid. Many 2-D and 3-D problems can be solved in 1-D equations if they have angular and radial symmetry. The reduced equations will typically involve geometric source terms. Therefore, conventional numerical schemes for Cartesian grid may not work well. We propose several approaches to apply the high order weighted essentially non-oscillatory (WENO) scheme to the 1D cylindrical and spherical grid. We have tested these schemes with Sedov explosion problem, and have found that the conservation in multi-dimensional sense is essential to generate physical solutions. The numerical results show that the global flux-splitting may fail to work even for high order WENO finite-difference schemes. We have also shown that only high order WENO finite-volume schemes can achieve both the high order accuracy and the conservation. Keyword: PDE, WENO, cylindrical and spherical, Euler equations, Sedov