An adaptive mesh-moving and local refinement method for time-dependent partial differential equations

Authors:
David C. Arney;Joseph E. Flaherty
Affiliations:
United States Military Academy, West Point, NY;Rensselaer Polytechnic Institute, Troy, NY
Venue:
ACM Transactions on Mathematical Software (TOMS)
Year:
1990

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Abstract

We discuss mesh-moving, static mesh-regeneration, and local mesh-refinement algorithms that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time-dependent partial differential equations in two space dimensions and time. A coarse base mesh of quadrilateral cells is moved by an algebraic mesh-movement function so as to follow and isolate spatially distinct phenomena. The local mesh-refinement method recursively divides the time step and spatial cells of the moving base mesh in regions where error indicators are high until a prescribed tolerance is satisfied. The static mesh-regeneration procedure is used to create a new base mesh when the existing one becomes too distorted. The adaptive methods have been combined with a MacCormack finite difference scheme for hyperbolic systems and an error indicator based upon estimates of the local discretization error obtained by Richardson extrapolation. Results are presented for several computational examples.