https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-05.pdf

Well-balanced schemes for the Euler equations with gravitation

We present a novel well-balanced second order Godunov-type finite volume scheme for compressible Euler equations with gravity. The well-balanced property is achieved by a specific combination of source term discretization, hydrostatic reconstruction, and numerical flux that exactly resolves stationary contacts. The scheme is able to preserve isothermal and polytropic stationary solutions up to machine precision. It is applied on several examples using the numerical flux of Roe to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.

A Second Order Well-Balanced Finite Volume Scheme for Euler Equations with Gravity

Well-balanced schemes for the Euler equations with gravitation: Conservative formulation using global fluxes

Context. Many problems in astrophysics feature flows which are close to hydrostatic equilibrium. However, standard numerical schemes for compressible hydrodynamics may be deficient in approximating this stationary state, where the pressure gradient is nearly balanced by gravitational forces.Aims. We aim to develop a second-order well-balanced scheme for the Euler equations. The scheme is designed to mimic a discrete version of the hydrostatic balance. It therefore can resolve a discrete hydrostatic equilibrium exactly (up to machine precision) and propagate perturbations, on top of this equilibrium, very accurately.Methods. A local second-order hydrostatic equilibrium preserving pressure reconstruction is developed. Combined with a standard central gravitational source term discretization and numerical fluxes that resolve stationary contact discontinuities exactly, the well-balanced property is achieved.Results. The resulting well-balanced scheme is robust and simple enough to be very easily implemented within any existing computer code that solves time explicitly or implicitly the compressible hydrodynamics equations. We demonstrate the performance of the well-balanced scheme for several astrophysically relevant applications: wave propagation in stellar atmospheres, a toy model for core-collapse supernovae, convection in carbon shell burning, and a realistic proto-neutron star.

/pdf/a-well-balanced-finite-volume-scheme-for-the-euler-equations-1glu7os7zz.pdf

A well-balanced finite volume scheme for the Euler equations with gravitation - The exact preservation of hydrostatic equilibrium with arbitrary entropy stratification

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

Summary
This paper describes a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear partial differential equation, whose solutions are called hydrostatic equilibria. We present a well-balanced method, meaning that besides discretizing the complete equations, the method is also able to maintain all hydrostatic equilibria. The method is a finite volume method, whose Riemann solver is approximated by a so-called relaxation Riemann solution that takes all hydrostatic equilibria into account. Relaxation ensures robustness, accuracy, and stability of our method, because it satisfies discrete entropy inequalities. We will present numerical examples, illustrating that our method works as promised. Copyright © 2015 John Wiley & Sons, Ltd.

A well-balanced scheme to capture non-explicit steady states in the Euler equations with gravity

The present paper concerns the derivation of numerical schemes to approximate the weak solutions of the Ripa model, which is an extension of the shallow-water model where a gradient of temperature is considered. Here, the main motivation lies in the exact capture of the steady states involved in the model. Because of the temperature gradient, the steady states at rest, of prime importance from the physical point of view, turn out to be very nonlinear and their exact capture by a numerical scheme is very challenging. We propose a relaxation technique to derive the required scheme. In fact, we exhibit an approximate Riemann solver that satisfies all the needed properties (robustness and well-balancing). We show three relaxation strategies to get a suitable interpretation of this adopted approximate Riemann solver. The resulting relaxation scheme is proved to be positive preserving, entropy satisfying and to exactly capture the nonlinear steady states at rest. Several numerical experiments illustrate the relevance of the method.

/pdf/well-balanced-schemes-to-capture-non-explicit-steady-states-108dgupanq.pdf

Well-balanced schemes to capture non-explicit steady states: Ripa model

We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss---Lobatto---Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.

Well-Balanced Nodal Discontinuous Galerkin Method for Euler Equations with Gravity

The aim of this work is to derive a well-balanced numerical scheme to approximate the solutions of the Euler equations with a gravitational potential. This system admits an infinity of steady state solutions which are not all known in an explicit way. Among all these solutions, the hydrostatic atmosphere has a special physical interest. We develop an approximate Riemann solver using the formalism of Harten, Lax and van Leer, which takes into account the source term. The resulting numerical scheme is proven to be robust, to preserve exactly the hydrostatic atmosphere and to preserve an approximation of all the other steady state solutions.

A Well-Balanced Scheme for the Euler Equation with a Gravitational Potential

We present a well-balanced finite volume solver for the compressible Euler equations with gravity where the approximate Riemann solver is derived using a relaxation approach. Besides the well-balanced property, the scheme is robust with respect to the physical admissible states. Another feature of the method is that it can maintain general stationary solutions of the hydrostatic equilibrium up to machine precision. For the first order scheme we present a well-balanced and positivity preserving second order extension using a modified minmod slope limiter. To maintain the well-balanced property, we reconstruct in equilibrium variables. Numerical examples are performed to demonstrate the accuracy, well-balanced and positivity preserving property of the presented scheme for up to 3 space dimensions.

Markus Zenk

Papers

A well-balanced scheme to capture non-explicit steady states in the Euler equations with gravity

Well-balanced schemes to capture non-explicit steady states: Ripa model

Well-Balanced Nodal Discontinuous Galerkin Method for Euler Equations with Gravity

A Well-Balanced Scheme for the Euler Equation with a Gravitational Potential

A second order positivity preserving well-balanced finite volume scheme for Euler equations with gravity for arbitrary hydrostatic equilibria