A discrete operator calculus for finite difference approximations

In this article we describe two areas of recent progress in the constructio n of accurate and robust finite difference algorithms for continuum dynamic s. The support operators method (SOM) provides a conceptual framework for d eriving a discrete operator calculus, based on mimicking selected propertie s of the differential operators. In this paper, we choose to preserve the f undamental conservation laws of a continuum in the discretization. A streng th of SOM is its applicability to irregular unstructured meshes. We describ e the construction of an operator calculus suitable for gas dynamics and fo r solid dynamics, derive general formulae for the operators, and exhibit th eir realization in 2D cylindrical coordinates. The multidimensional positiv e 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-la grangian-Eulerian (ALE) programs. MPDATA is based on the sign-preserving pr operty 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 al so to a complex fluid flow. We also provide an example to demonstrate the b lending of the SOM and MPDATA approaches. (C) 2000 Elsevier Science S.A. Al l rights reserved.