Positivity preserving and entropy consistent approximate Riemann solvers dedicated to the high-order MOOD-based Finite Volume discretization of Lagrangian and Eulerian gas dynamics

2021 ◽  
pp. 105056
Author(s):  
Agnes Chan ◽  
Gérard Gallice ◽  
Raphaël Loubère ◽  
Pierre-Henri Maire
Keyword(s):  
Finite Volume ◽  
Gas Dynamics ◽  
High Order ◽  
Positivity Preserving ◽  
Riemann Solvers ◽  
Finite Volume Discretization ◽  
Approximate Riemann Solvers

scholarly journals Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

2011 ◽  
Vol 9 (2) ◽  
pp. 324-362 ◽  
Author(s):  
Franz Georg Fuchs ◽  
Andrew D. McMurry ◽  
Siddhartha Mishra ◽  
Nils Henrik Risebro ◽  
Knut Waagan
Keyword(s):  
Finite Volume ◽  
Source Term ◽  
Second Order ◽  
High Order ◽  
Mhd Equations ◽  
Positive Pressure ◽  
Finite Volume Schemes ◽  
Riemann Solvers ◽  
Ideal Mhd ◽  
Approximate Riemann Solvers

AbstractWe design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

AUSM-Based High-Order Solution for Euler Equations

2012 ◽  
Vol 12 (4) ◽  
pp. 1096-1120 ◽  
Author(s):  
Angelo L. Scandaliato ◽  
Meng-Sing Liou
Keyword(s):  
Euler Equations ◽  
Splitting Method ◽  
High Order ◽  
Riemann Solvers ◽  
Carbuncle Phenomenon ◽  
Solution Accuracy ◽  
Matrix Vector ◽  
Approximate Riemann Solvers ◽  
Monotonicity Preserving ◽  
High Order Interpolation

AbstractIn this paper we demonstrate the accuracy and robustness of combining the advection upwind splitting method (AUSM), specifically AUSM+-UP, with high-order upwind-biased interpolation procedures, the weighted essentially non-oscillatory (WENO-JS) scheme and its variations, and the monotonicity preserving (MP) scheme, for solving the Euler equations. MP is found to be more effective than the three WENO variations studied. AUSM+-UP is also shown to be free of the so-called “carbuncle” phenomenon with the high-order interpolation. The characteristic variables are preferred for interpolation after comparing the results using primitive and conservative variables, even though they require additional matrix-vector operations. Results using the Roe flux with an entropy fix and the Lax-Friedrichs approximate Riemann solvers are also included for comparison. In addition, four reflective boundary condition implementations are compared for their effects on residual convergence and solution accuracy. Finally, a measure for quantifying the efficiency of obtaining high order solutions is proposed; the measure reveals that a maximum return is reached after which no improvement in accuracy is possible for a given grid size.

scholarly journals Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

2011 ◽  
Vol 467 (2134) ◽  
pp. 2752-2776 ◽  
Author(s):  
Xiangxiong Zhang ◽  
Chi-Wang Shu
Keyword(s):  
Maximum Principle ◽  
Conservation Laws ◽  
Finite Volume ◽  
Computational Cost ◽  
High Order ◽  
High Order Accuracy ◽  
Boltzmann Transport ◽  
Finite Volume Schemes ◽  
Positivity Preserving ◽  
High Order Schemes

In an earlier study (Zhang & Shu 2010 b J. Comput. Phys. 229 , 3091–3120 ( doi:10.1016/j.jcp.2009.12.030 )), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensions for arbitrarily high-order schemes on structured and unstructured meshes. The same idea can be used to construct high-order schemes preserving the positivity of certain physical quantities, such as density and pressure for compressible Euler equations, water height for shallow water equations and density for Vlasov–Boltzmann transport equations. These schemes have been applied in computational fluid dynamics, computational astronomy and astrophysics, plasma simulation, population models and traffic flow models. In this paper, we first review the main ideas of these maximum-principle-satisfying and positivity-preserving high-order schemes, then present a simpler implementation which will result in a significant reduction of computational cost especially for weighted essentially non-oscillatory finite-volume schemes.

scholarly journals High-order methods for the simulation of hydromagnetic instabilities in core-collapse supernovae

2010 ◽  
Vol 6 (S274) ◽  
pp. 479-481
Author(s):  
T. Rembiasz ◽  
M. Obergaulinger ◽  
M. Angel Aloy ◽  
P. Cerdá-Durán ◽  
E. Müller
Keyword(s):  
High Order ◽  
Core Collapse ◽  
High Order Methods ◽  
Riemann Solvers ◽  
Tearing Instability ◽  
Constrained Transport ◽  
Approximate Riemann Solvers

AbstractWe present an assessment of the accuracy of a recently developed MHD code used to study hydromagnetic flows in supernovae and related events. The code, based on the constrained transport formulation, incorporates unprecedented ultra-high-order methods (up to 9th order) for the reconstruction and the most accurate approximate Riemann solvers. We estimate the numerical resistivity of these schemes in tearing instability simulations.

Sign in / Sign up

Export Citation Format

Share Document