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 class of schemes: the Residual Distribution one \cite{MR4090481}, and the Active Flux formulations \cite{AF1, AF3, AF4,AF5,RoeAF}. 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 pointwise and cella averaged degrees of freedom is different, and hence follow different form 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.
In this paper, we study both convergence and bounded variation properties of a new fully discrete conservative Lagrangian--Eulerian scheme to the entropy solution in the sense of Kruzhkov (scalar case) by using a weak asymptotic analysis. We discuss theoretical developments on the conception of no-flow curves for hyperbolic problems within scientific computing. The resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions. We present experiments to a study based on the use of the Wasserstein distance to show the effectiveness of the n…
In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine-Hugoniot junmp condition. Unfortunately, to avoid the …
Testing the order of accuracy of (very) high order methods for shallow water (and Euler) equations is a delicate operation and the test cases are the crucial starting point of this operation. We provide a short derivation of vortex-like analytical solutions in 2 dimensions for the shallow water equations (and, hence, Euler equations) that can be used to test the order of accuracy of numerical methods. These solutions have different smoothness in their derivatives (up to $\mathcal C^\infty$) and can be used accordingly to the order of accuracy of the scheme to test.
Arbitrary order stationarity preserving stabilized finite elements for multidimensional nonlinear hyperbolic problems. Application to the Euler equations with gravity
Dip. di Matematica G. Castelnuovo, Università di Roma La Sapienza（罗马第一大学数学G.卡斯特鲁诺沃系）；LaR2A Laboratory, Dep. of Mathematics, Faculty of Sciences, Abdelmalek Essaadi Univ.（阿卜杜勒马利克·埃萨迪大学理学院数学系LaR2A实验室）；Inria, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（法国国家信息与自动化研究所、波尔多大学、法国国家科学研究中心、波尔多理工学院、IMB）
Use of Jordan forms for convection-pressure split Euler solvers
TIFR Center for Applicable Mathematics（塔塔基础研究所应用数学中心）；Indian Institute of Science (IISc)（印度理工学院）；Department of Aerospace Engineering, IISc（印度理工学院航空航天工程系）；Department of Civil Engineering, IISc（印度理工学院土木工程系）
The design and analysis of a unified asymptotic preserving (AP) and well-balanced scheme for the Euler Equations with gravitational and frictional source terms is presented in this paper. The asymptotic behaviour of the Euler system in the limit of zero Mach and Froude numbers, and large friction is characterised by an additional scaling parameter. Depending on the values of this parameter, the Euler system relaxes towards a hyperbolic or a parabolic limit equation. Standard Implicit-Explicit Runge-Kutta schemes are incapable of switching between these asymptotic regimes. We propose a time sem…
The paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for $n\times n$ hyperbolic conservation laws in one space dimension. These estimates are achieved by a "post-processing algorithm", checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained by the Godunov or the Lax-Friedrichs scheme, backward Euler approximations, and the method of periodic smoothing. Some numerical implementations are presented.
We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation. The MOC gives rise to two mutually coupled systems of ordinary differential equations. As a special case we consider the Monge-Ampère equation, for which we solve the system of ODE's using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Numerical examples demonstrate the performance of the methods.
This paper describes a class of scheme named "residual distribution schemes" or "fluctuation splitting schemes". They are a generalization of Roe's numerical flux in fluctuation form. The so-called multidimensional fluctuation schemes have historically first been developed for steady homogeneous hyperbolic systems. Their application to unsteady problems and conservation laws has been really understood only relatively recently. This understanding has allowed to make of the residual distribution framework a powerful playground to develop numerical discretizations embedding some prescribed constr…
Hyperbolic partial differential equations (PDEs) cover a wide range of interesting phenomena, from human and hearth-sciences up to astrophysics: this unavoidably requires the treatment of many space and time scales in order to describe at the same time observer-size macrostructures, multi-scale turbulent features, and also zero-scale shocks. Moreover, numerical methods for solving hyperbolic PDEs must reliably handle different families of waves: smooth rarefactions, and discontinuities of shock and contact type. In order to achieve these goals, an effective approach consists in the combination…
Intrusive Uncertainty Quantification methods such as stochastic Galerkin are gaining popularity, whereas the classical stochastic Galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system. We apply a modification of this method that uses a slope limiter to retain admissible solutions of the system, while providing high-order approximations in the physical and stochastic space. This is done using a spatial discontinuous Galerkin scheme and a Multi-Element stochastic Galerkin ansatz in the random space. We analyze the convergence of the resulting scheme and a…
We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical method is the presence of shock waves that are resolved by proposing a numerical discretization of the calculus introduced in Bressan and Marson [Rend. Sem. Mat. Univ. Padova, 94:79-94, 1995]. Numerical results are presented for the one-dimensional Burgers equation and the Euler equations. Using the essential routines of a state-of-the-art code for computationa…
High-order discontinuous Galerkin schemes with subcell finite volume limiter and adaptive mesh refinement for a monolithic first-order BSSNOK formulation of the Einstein-Euler equations
