High Order Multi-Moment Finite Volume Method and the Applications to Fluid Dynamics

Abstract

High-order finite volume methods (FVM) have been developed using the underlying idea of the CIP (Constrained Interpolation Profile) method. In addition to the volume-integrated average (VIA) that is the basic variable in conventional FVM, other kinds of quantities, such as point value (PV) and derivative (DV) of a physical field, which are generically called “moments” in our context, are also treated as the model variables and carried forward in time independently. With all these moments readily updated at every time step, high-order reconstruction can be built on a compact grid stencil. In fact, only a single cell is required in almost all cases. The way to choose and update the different moments is quite flexible. For example, the PV and DV can be computed by either the semi-Lagrangian approach or solved locally as derivative Riemann problems. The VIA moment, however, has to be computed by a flux-based finite volume formulation that ensures exactly the numerical conservativeness. We have explored several numerical formulations of this sort for practical use. The proposed schemes are well suited for unstructured grids and adaptive refinement techniques. More numerical details and applications in fluid dynamics will be reported in the conference.