No 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.

It is well known, thanks to Lax-Wendroff theorem, that the local conservation of a numerical scheme for a conservative hyperbolic system is a simple and systematic way to guarantee that, if stable, a scheme will provide a sequence of solutions that will converge to a weak solution of the continuous problem. In [1], it is shown that a nonconservative scheme will not provide a good solution. The question of using, nevertheless, a nonconservative formulation of the system and getting the correct solution has been a long-standing debate. In this paper, we show how to get a relevant weak solution from a pressure-based formulation of the Euler equations of fluid mechanics. This is useful when dealing with nonlinear equations of state because it is easier to compute the internal energy from the pressure than the opposite. This makes it possible to get oscillation-free solutions, contrarily to classical conservative methods. An extension to multiphase flows is also discussed.

Abstract. We consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen-Loève expansion on the state space of the random flux. For random flux functions which are continuosly differentiable with respect to the state variable u, we prove the existence of a unique random entropy solution. Using a Karhunen-Loève spectral decomposition of the random flux into principal components with respect to the state variables, we introduce a family of parametric, deterministic entropy solutions on high-dimensional parameter spaces. We prove bounds on the sensitivity of the parametric and of the random entropy solutions on the Karhunen-Loève parameters. We also outline the convergence analysis for two classes of discretization schemes, the Multi-Level Monte-Carlo Finite-Volume Method (MLMCFVM) developed in [21,23,22], and the stochastic collocation Finite Volume Method (SCFVM) of [29].

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.