A discrete operator calculus for finite difference approximations

Abstract

In this article we describe two areas of recent progress in the construction of accurate and robust finite difference algorithms for continuum dynamics. The support operators method (SOM) provides a conceptual framework for deriving a discrete operator calculus, based on mimicking selected properties of the differential operators. In this paper, we choose to preserve the fundamental conservation laws of a continuum in the discretization. A strength of SOM is its applicability to irregular unstructured meshes. We describe the construction of an operator calculus suitable for gas dynamics and for solid dynamics, derive general formulae for the operators, and exhibit their realization in 2D cylindrical coordinates. The multidimensional positive definite advection transport algorithm (MPDATA) provides a framework for constructing accurate nonoscillatory advection schemes. In particular, the nonoscillatory property is important in the remapping stage of arbitrary-Lagrangian-Eulerian (ALE) programs. MPDATA is based on the sign-preserving property of upstream differencing, and is fully multidimensional. We describe the basic second-order-accurate method, and review its generalizations. We show examples of the application of MPDATA to an advection problem, and also to a complex fluid flow. We also provide an example to demonstrate the blending of the SOM and MPDATA approaches.