scholarly journals Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system

2016 ◽  
Vol 26 (13) ◽  
pp. 2525-2615 ◽  
Author(s):  
S. Dellacherie ◽  
J. Jung ◽  
P. Omnes ◽  
P.-A. Raviart
Keyword(s):  
Wave Equation ◽  
Mach Number ◽  
Stability Result ◽  
Linear Analysis ◽  
Euler System ◽  
Linear Wave ◽  
Godunov Scheme ◽  
Linear Wave Equation ◽  
Number Problem ◽  
Theoretical Results

This paper is composed of three self-consistent sections that can be read independently of each other. In Sec. 1, we define and analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov-type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Sec. 2, we apply the all Mach correction proposed in Sec. 1 to the case of the nonlinear barotropic Euler system when the Godunov-type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this nonlinear case by showing how this construction is related with the linear analysis of Sec. 1. At last, we apply in Sec. 3 the all Mach correction proposed in Sec. 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Secs. 1–3 justify the theoretical results and show that the obtained all Mach Godunov-type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

scholarly journals Parameter identification for the linear wave equation with Robin boundary condition

2019 ◽  
Vol 27 (1) ◽  
pp. 25-41
Author(s):  
Valeria Bacchelli ◽  
Dario Pierotti ◽  
Stefano Micheletti ◽  
Simona Perotto
Keyword(s):  
Wave Equation ◽  
Mixed Boundary ◽  
Stability Result ◽  
Interval Length ◽  
Left Endpoint ◽  
Linear Wave ◽  
Reconstruction Procedure ◽  
Element Discretization ◽  
Linear Wave Equation ◽  
Bounded Interval

Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

scholarly journals Construction of a low Mach finite volume scheme for the isentropic Euler system with porosity

2021 ◽  
Author(s):  
Pascal Omnes ◽  
Stéphane Dellacherie ◽  
Jonathan Jung
Keyword(s):  
Mach Number ◽  
Finite Volume ◽  
Euler System ◽  
Finite Volume Scheme ◽  
Linear Wave ◽  
Finite Volume Schemes ◽  
Linear Wave Equation ◽  
Continuous Space ◽  
Low Mach Number ◽  
Non Linear System

Classical finite volume schemes for the Euler system  are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.

The influence of cell geometry on the Godunov scheme applied to the linear wave equation

2010 ◽  
Vol 229 (14) ◽  
pp. 5315-5338 ◽  
Author(s):  
Stéphane Dellacherie ◽  
Pascal Omnes ◽  
Felix Rieper
Keyword(s):  
Wave Equation ◽  
Linear Wave ◽  
Godunov Scheme ◽  
Cell Geometry ◽  
Linear Wave Equation

scholarly journals Energy decay rate for a quasi-linear wave equation with localized strong dissipation

2011 ◽  
Vol 62 (1) ◽  
pp. 164-172 ◽  
Author(s):  
Daewook Kim ◽  
Yong Han Kang ◽  
Mi Jin Lee ◽  
Il Hyo Jung
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Energy Decay ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Energy Decay Rate

scholarly journals Godunov type scheme for the linear wave equation with Coriolis source term

2017 ◽  
Vol 58 ◽  
pp. 1-26 ◽  
Author(s):  
Emmanuel Audusse ◽  
Stéphane Dellacherie ◽  
Minh Hieu Do ◽  
Pascal Omnes ◽  
Yohan Penel
Keyword(s):  
Wave Equation ◽  
Source Term ◽  
Linear Wave ◽  
Linear Wave Equation

scholarly journals Radial solutions to the Cauchy problem for □1+3U = 0 as limits of exterior solutions

2016 ◽  
Vol 13 (04) ◽  
pp. 833-860
Author(s):  
Helge Kristian Jenssen ◽  
Charis Tsikkou
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Cauchy Data ◽  
Linear Wave ◽  
Radial Solutions ◽  
Linear Wave Equation ◽  
Model Situation ◽  
Neumann Solutions

We consider the strategy of realizing the solution of a Cauchy problem (CP) with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the three-dimensional (3d), linear wave equation [Formula: see text] with radial Cauchy data [Formula: see text], [Formula: see text]. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric CP. In treating the 3d wave equation, we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in [Formula: see text] whenever [Formula: see text]. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in [Formula: see text], we obtain [Formula: see text]-solutions via exterior Neumann solutions, and only [Formula: see text]-solutions via exterior Dirichlet solutions.

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