A combination of residual distribution and the active flux formulations or a new class of schemes that can combine several writings of the same hyperbolic problem: Application to the 1D Euler equations

We show how to combine in a natural way (i.e., without any test nor switch) the conservative and non-conservative formulations of an hyperbolic system that has a conservative form. This is inspired from two different classes of schemes: the residual distribution one (Abgrall in Commun Appl Math Comput 2(3): 341–368, 2020), and the active flux formulations (Eyman and Roe in 49th AIAA Aerospace Science Meeting, 2011; Eyman in active flux. PhD thesis, University of Michigan, 2013; Helzel et al. in J Sci Comput 80(3): 35–61, 2019; Barsukow in J Sci Comput 86(1): paper No. 3, 34, 2021; Roe in J Sci Comput 73: 1094–1114, 2017). The solution is globally continuous, and as in the active flux method, described by a combination of point values and average values. Unlike the “classical” active flux methods, the meaning of the point-wise and cell average degrees of freedom is different, and hence follow different forms of PDEs; it is a conservative version of the cell average, and a possibly non-conservative one for the points. This new class of scheme is proved to satisfy a Lax-Wendroff-like theorem. We also develop a method to perform non-linear stability. We illustrate the behaviour on several benchmarks, some quite challenging

Multidimensional upwind hydrodynamics on unstructured meshes using graphics processing units - I. Two-dimensional uniform meshes

SJP is supported by a Royal Society University Research Fellowship

A multidimensional positive definite remapping for Lagrangian solutions of the Noh problem

A remapping based on the multidimensional positive definite advection transport algorithm (MPDATA), implemented for ALE methods, is used to model the Noh problem. Typical solutions in the Lagrangian reference frame contain heating errors which arise during the simulation of a shock reflection originating at a wall. The paper shows that the inherent properties of MPDATA can be exploited in the remapping to reduce wall heating errors. The resulting increase in accuracy and symmetry of solutions is demonstrated

Multi-Dimensional Astrophysical Structural and Dynamical Analysis I. Development of a Nonlinear Finite Element Approach

A new field of numerical astrophysics is introduced which addresses the solution of large, multidimensional structural or slowly-evolving problems (rotating stars, interacting binaries, thick advective accretion disks, four dimensional spacetimes, etc.). The technique employed is the Finite Element Method (FEM), commonly used to solve engineering structural problems. The approach developed herein has the following key features: 1. The computational mesh can extend into the time dimension, as well as space, perhaps only a few cells, or throughout spacetime. 2. Virtually all equations describing the astrophysics of continuous media, including the field equations, can be written in a compact form similar to that routinely solved by most engineering finite element codes. 3. The transformations that occur naturally in the four-dimensional FEM possess both coordinate and boost features, such that (a) although the computational mesh may have a complex, non-analytic, curvilinear structure, the physical equations still can be written in a simple coordinate system independent of the mesh geometry. (b) if the mesh has a complex flow velocity with respect to coordinate space, the transformations will form the proper arbitrary Lagrangian- Eulerian advective derivatives automatically. 4. The complex difference equations on the arbitrary curvilinear grid are generated automatically from encoded differential equations. This first paper concentrates on developing a robust and widely-applicable set of techniques using the nonlinear FEM and presents some examples.Comment: 28 pages, 9 figures; added integral boundary conditions, allowing very rapidly-rotating stars; accepted for publication in Ap.