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A discrete operator calculus for finite difference approximations. (English) Zbl 0978.76063
Summary: 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 convervation 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 examine their realization in 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 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 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.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography
Software:
MPDATA
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