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

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.

Adaptive Methods and Parallel Computation for Partial Differential Equations

“…Previous investigations [4,6] revealed that error estimates were generally better than 80 percent of the actual error for a wide range of mesh spacings and problems. Equation (3) can be used to select refinement factors other than binary and, indeed, to select different refinement levels for different meshes at a given tree level.…”

“…Adaptive strategies in current practice are classified as h-, p-, or rrefinement when, respectively, computational meshes are refined or coarsened in regions of the problem domain that require more or less resolution [6,12], the order of accuracy is varied in different regions [101, or a fixed-topology mesh is redistributed [5]. These basic enrichment methods may be used alone or in combination.…”

“…Our research is based on a serial adaptive hr-refinement algorithm of Arney and Flaherty [6]. We forego mesh motion at present and briefly describe an h-refinement procedure that utilizes their strategy.…”

“…Following Arney et al [4,6], refinement indicators are selected as estimates of the local discretization error obtained by Richardson's extrapolation (h-refinement) on a mesh having half the spatial and temporal spacing of the mesh used to generate the solution. Fine-mesh solutions generated as part of this error estimation process may subsequently be used on finer meshes when refinement is necessary.…”

Adaptive Mesh Refinement for Computational Aeroacoustics

“…With this method there was an extensive improvement of computational efficiency for solving shock hydrodynamic problems [13]. Combined this AMR method with an adaptive mesh-moving algorithm a time-dependent problem of moving shock along a flat wall was simulated efficiently too [43]. It was also extended to solving applications governed by the Navier-Stokes equations [44].…”