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.