ReALE - Reconnection-based Multimaterial Arbitrary Lagrangian-Eulerian Method Mimetic Methods for Partial Differential Equations

The need of the office of science programs to approximate the solutions of strongly nonlinear,coupled partial differential equations in complex domains has been a continuous driver for the dualdevelopment of supercomputing platforms and for more accurate and efficient numerical algorithms.Despite the years and magnitude of the effort that has been put into computational science, in manyways the construction of new algorithms for complex problems remains as much of an art than ascience. While the accuracy and efficiency of an algorithms for idealized problems can be studiedwith mathematical tools of numerical analysis, increased predictiveness of realistic simulations moretypically requires the incorporation of physical principles into the algorithm. The goal of ”MimeticMethods for Partial Differential Equations” project at LANL (M. Shashkov, PI) is to developmethods that are constructed to embed physical principles.The physical principles that might beincorporated include causality, coordinate invariance, conservation laws, symmetries, asymptotics,and well-posedness. In first part of the talk I will give brief overview of the project and describemain recent achievements. It includes discretization of diffusion and Maxwell’s equations on 3Dgeneralized polyhedral meshes with non-flat faces; enforcement discrete maximum principle fordiffusion and advection-diffusion equations mimetic discretization of equations of linear elasticity,high-order discretizations for diffusion problem, remapping and closure models for multi-materialarbitrary Lagrangian-Eulerian (ALE) methods.In second part of my talk I will describe new reconnection-based arbitrary Lagrangian-Eulerian(ReALE) method for high-speed compressible multi-material flows. This work done in collabo-ration with R. Loubere, P.-H. Maire, J. Breil, S. Galera. ReALE method for can be viewed asan extended ALE methodology that allows changes of topology at rezone stage [1,2]. Anotherimportant difference compared to classical ALE lies in the way rezoning is performed. Indeed,by adding cell-based moving generators, one constructs a Voronoi tessellation at each time step(rezoned mesh). A quasi-Lagrangian motion of generators allows the preservation of the accuracyof the Lagrangian scheme. On the other hand, a quasi-centroidal motion allows to build a smoothrezoned mesh. We developed an intermediate Lagrangian/centroidal motion based on a weightfunction derived from eigenvalues of the one time step deformation tensor. The ability for gen-erators to dynamically change neighborhood enables reconnection; when vorticity occurs this is aparticularly attractive feature. An exact intersection-based remapper is then used to conservativelyproject physical quantities from the Lagrangian mesh onto the new rezoned one (the reconnectedVoronoi mesh obtained after generators motion). We will present numerical evidence (which in-cludes several problems involving vorticity formation and presence of strong shear deformation,for example, bubble/shock interaction, Rayleigh-Taylor instability) that this reconnection friendlyframework drastically extends the capability of any classical ALE code. References[1] R. Loubere,et al., ReALE: A Reconnection-based Arbitrary-Lagrangian-Eulerian Method, J. of Comput. Phys., 229