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Abstract: Finite scale equations are coarse-grained PDEs that describe the evolution of density, momentum and energy fields averaged over finite intervals of space and time. These analytic equations have been found to be a useful model for analyzing and verifying discrete algorithms employed in numerical simulation. In this paper, we derive traveling wave solutions for finite scale shocks and compare the results with averaged solutions of Navier–Stokes. We find that the finite scale equations accurately predict the shock speed and width and the jump conditions relating the pre and post shock states. [Copyright &y& Elsevier]

An accurate remapping algorithm is an essential component of many arbitrary Lagrangian–Eulerian (ALE) methods. In previous work, we have described a local remapping algorithm for a positive cell-centered scalar function that is second-order accurate, conservative, and sign preserving. However remapping in the context of high speed flow introduces new issues, which include the consistent treatment of the kinetic and internal energies, compatible remapping of mass and momentum on the staggered mesh, and the generalization from sign preservation to monotonicity preservation. We describe a remap strategy that deals with each of these issues. Although the theoretical development of this strategy is intricate, the resulting scheme is both simple to implement and efficient. We provide numerical examples to illustrate the individual steps of the remap and its overall performance. [Copyright &y& Elsevier]

An accurate conservative interpolation (remapping) algorithm is an essential component of most arbitrary Lagrangian–Eulerian (ALE) methods. In this paper we describe a local remapping algorithm for a positive scalar function. This algorithm is second-order accurate, conservative, and sign preserving. The algorithm is based on estimating the mass exchanged between cells at their common interface, and so is equally applicable to structured and unstructured grids. We construct the algorithm in a series of steps, clearly delineating the assumptions and errors made at each step. We validate our theory with a suite of numerical examples, analyzing the results from the viewpoint of accuracy and order of convergence. [Copyright &y& Elsevier]

Analyzes the zero-viscosity Multi-dimensional Positive Definite Advection Transport Algorithm (MPDATA) results. Properties and implementation of MPDATA; Simulations of decaying turbulence of a homogenous incompressible fluid in a triply periodic cube; Summary of theoretical results.