A multidimensional positive definite remapping algorithmn for arbitrary Lagrangian-Eulerian methods

A second-order conservative sign-preserving remapping scheme for arbitrary Lagrangian–Eulerian (ALE) methods is developed utilising concepts of the multidimensional positive definite advection transport algorithm (MPDATA). The algorithm is inherently multidimensional, and so does not introduce splitting errors. The remapping is implemented in a two-dimensional, finite element ALE solver employing staggered quadrilateral meshes. The MPDATA remapping uses a finite volume discretisation developed for volume coordinates. It is applied for the remapping of density and internal energy arranged as cell centered, and velocity as nodal, dependent variables. The numerical investigations include an asymptotic mesh convergence study and comparisons of MPDATA with remapping based upon the van Leer MUSCL algorithm. Theoretical considerations are supported with examples involving idealised cases with prescribed mesh movement for advection of scalars, and single material ALE solutions for benchmarks of the Explosion and Noh problems. The latter illustrates an optional wall heating treatment naturally arising from the properties of MPDATA. The results demonstrate the advantages of fully multidimensional remapping, and show that the properties of MPDATA remapping are retained for fields with arbitrary sign