High-Order Finite-Volume Methods on Locally-Structured Grids

For many problems in astrophysics and space sciences, it is desirable to compute solutions in a way that preserves spherical symmetry, so that the dynamics of small perturbations about the spherically symmetric case are not overwhelmed by numerical error. Traditionally, such calculations have been done by discretizing the equations expressed in spherical coordinates. This approach has significant numerical problems, due to the singularity in the spherical coordinate system at the poles. An alternative approach is based on the “cubed-sphere” representation (Figure 1). In this approach, a solid sphere is represented as the disjoint union of the images of smooth mappings of multiple rectangular blocks. Away from the block at the center of the sphere, the coordinate surfaces of the mappings coincide with spherical surfaces, thus providing a coordinate system that, when discretized, will preserve spherical symmetry, at least away from the center of the sphere. At the block boundaries, the coordinate lines meet continuously, but are not smooth. In addition, the grids meet in a way that does not lead to a single logically rectangular coordinate transformation. These features are difficulties for standard finite-volume methods on structured grids. The accuracy of such methods depends strongly on the smoothness of the grid mapping. At places where the grid mapping fails to be smooth, there is a loss of one order of accuracy in the truncation error. For a second-order accurate method, this leads to a local truncation error that is first-order at block boundaries. Similar issues arise for finite-volume methods on locally-refined structured grids: at refinement boundaries, such methods lose one order of accuracy. For many problems, a such a reduction to first-order accuracy is unacceptable, even on a set of codimension one. To address this issue, we we are developing a new class of finite-volume methods on locallyrefined and mapped-multiblock grids. The central feature of these methods is that they are at least fourth-order accurate in regions where the solution is smooth; otherwise, we want to retain to as great an extent as possible the advantages of traditional finite-volume methods. In this paper, we discuss our approach to the design of such methods.