An arbitrary Lagrangian–Eulerian method with adaptive mesh refinement for the solution of the Euler equations

A new algorithm that combines staggered grid arbitrary Lagrangian–Eulerian (ALE) techniques with structured local adaptive mesh refinement (AMR) has been developed for the solution of the Euler equations. The novel components of the method are driven by the need to reconcile traditional AMR techniques with the staggered variables and moving, deforming meshes associated with Lagrange based ALE schemes. Interlevel coupling is achieved with refinement and coarsening operators, as well as mesh motion boundary conditions. Elliptic mesh relaxation schemes are extended for use within the context of an adaptive mesh hierarchy. Numerical examples are used to highlight the utility of the method over single level ALE solution methods, facilitating substantial efficiency improvements and enabling the efficient solution of a highly resolved three-dimensional Richtmyer–Meshkov instability problem.