Adaptive High-order Finite Volume Discretizations on Spherical Thin Shells
Seminar Room 1, Newton Institute
We present an adaptive, conservative finite volume approach applicable to solving hyperbolic PDE's on both 2D surface and 3D thin shells on the sphere. The starting point for this method is the equiangular cubed-sphere mapping, which maps six rectangular coordinate patches (blocks) onto the sphere. The images of these blocks form a disjoint union covering the sphere, with the mappings of adjacent blocks being continuous, but not differentiable, at block boundaries. Our method uses a fourth-order accurate discretization to compute flux averages on faces, with a higher-order least squares-based interpolation to compute stencil operations near block boundaries. To suppress oscillations at discontinuities and underresolved gradients, we use a limiter that preserves fourth-order accuracy at smooth extrema, and a redistribution scheme to preserve positivity where appropriate for advected scalars. By using both space- and time-adaptive mesh refinement, the solver allocates comp utational effort only where greater accuracy is needed. The resulting method is demonstrated to be fourth-order accurate for advection and shallow water equation model problems, and robust at solution discontinuities. We will also present an approach for the compressible Euler equations on a 3D thin spherical shell. Refinement is performed only in the horizontal directions, The radial direction is treated implicitly (using a fourth-order RK IMEX scheme) to eliminate time step constraints from vertical acoustic waves.