Algebraic Flux Correction and Geometric Conservation in ALE Computations

Abstract

In this chapter, we describe the important role played by the so-called Geometric Conservation Law (GCL) in the design of Flux-Corrected Transport (FCT) methods for Arbitrary Lagrangian-Eulerian (ALE) applications. We propose a conservative synchronized remap algorithm applicable to arbitrary Lagrangian-Eulerian computations with nodal finite elements. Unique to the proposed method is the direct incorporation of the geometric conservation law (GCL) in the resulting numerical scheme. We show how the geometric conservation law allows the proposed method to inherit the positivity preserving and local extrema diminishing (LED) properties typical of FCT schemes for pure transport problems. The extension to systems of equations which typically arise in meteorological and compressible flow computations is performed by means of a synchronized strategy. The proposed approach also complements and extends the work of the first author on nodal-based methods for shock hydrodynamics, delivering a fully integrated suite of Lagrangian/remap algorithms for computations of compressible materials under extreme load conditions. Numerical tests in multiple dimensions show that the method is robust and accurate in typical computational scenarios.