High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics

Abstract

In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique, while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ODE solver in a classical finite element approach to advance the solution in time. In this work, we propose a new Residual Distribution (RD) scheme, which provides an arbitrary explicit high order approximation of the smooth solutions of the Euler equations both in space and time. The design of the scheme via the coupling of the RD formulation [1, 2] with a Deferred Correction (DeC) type method [3,4], allows to have the matrix associated to the update in time, which needs to be inverted, to be diagonal. The use of Bernstein polynomials as shape functions, guarantees that this diagonal matrix is invertible and ensures strict positivity of the resulting diagonal matrix coefficients. This work is the extension of [5, 6] to multidimensional systems. We have assessed our method on several challenging benchmark problems for one-and two-dimensional Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions. polynomials are a suitable choice, but this is not the only possible one. The idea to use as shape functions the Bernstein polynomials, instead of the more typical Lagrange polynomials, has been discussed in [7,6] applied to the context of high order residual distribution schemes and very recently, in [8], this idea has been applied for a different class of methods, namely, the flux-corrected transport method.The purpose of this paper is to show how these ideas can be further extended for solving the Euler equations of fluid dynamics for the simulation of flows involving strong discontinuities. The RD formulation used here is based on the finite element approximation of the solution as a globally continuous piecewise polynomial. The design principle of the new RD scheme guarantees a compact approximation stencil even for high order accuracy, which would hold for Discontinuous Galerkin (DG), but not for example for Finite Volume (FV) methods and allows to consider a smaller number of nodes than DG ([9, 10, 11]).The format of this paper is the following. In Section 2, we recall the idea of the residual distribution schemes for steady problems and in Section 3 we describe the time-stepping algorithm and adapt the method developed in [5,6] to multidimensional systems. We illustrate the robustness and accuracy of the proposed method by means of rigorous numerical tests and discuss the obtained results in Section 4. Finally, we give the conclusive remarks and outline further perspectives.

High Order Asymptotic Preserving Deferred Correction Implicit-Explicit Schemes for Kinetic Models

“…We use the residual distribution (RD) framework [3,6,15,24] to discretize our space. This class of schemes is a generalization of finite element methods (FEM); they use compact stencils, they do not need Riemann solvers, and they are easily generalizable.…”

“…Details and some examples of the schemes can be found in Appendix A. In particular, we will use the residual distributions, and hence the schemes, defined and tested in [6].…”

“…In this section, we will introduce the explicit DeC algorithm of [4,6]. This is a preliminary step to understand the relative IMEX version that we will present in section 5.…”

Neural Network-Based Limiter with Transfer Learning

“…Furthermore, we want to explore different strategies which can be used to perform the transfer learning. We describe how to adapt a neural network shock-indicator that has been trained on data from a modal DG scheme on a Cartesian mesh to a neural network shock-indicator that works on a residual distribution (RD) scheme ( [1,2,32,33] for a brief introduction) on a Cartesian mesh and an unstructured triangular mesh.…”

Multidimensional Staggered Grid Residual Distribution Scheme for Lagrangian Hydrodynamics

“…Next, in Section 4, we recall the RD formulation for time-dependent problems. In Section 4.2, we explain the diagonalization of the global sparse mass matrix without the loss of accuracy: this is obtained by modifying the timestepping method by applying ideas coming from [28,3,8,5,4]. In Section 5, we explain how to adapt the RD framework to the equations of Lagrangian hydrodynamics.…”