High Order Well-balanced WENO Scheme for the Gas Dynamics Equations under Gravitational Fields

Abstract

The gas dynamics equations, coupled with a static gravitational field, admit the hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term. Many astrophysical problems involve the hydrodynamical evolution in a gravitational field, therefore it is essential to correctly capture the effect of gravitational force in the simulations. Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run. In this paper we design high order well-balanced finite difference WENO schemes to this system, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy. Numerical tests are performed to verify high order accuracy, well-balanced property, and good resolution for smooth and discontinuous solutions. Keywords TR-2011-279. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited SUPPLEMENTARY NOTES ABSTRACTThe gas dynamics equations, coupled with a static gravitational field, admit the hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term. Many astrophysical problems involve the hydrodynamical evolution in a gravitational field, therefore it is essential to correctly capture the effect of gravitational force in the simulations. Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run. In this paper we design high order well-balanced finite difference WENO schemes to this system, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy. Numerical tests are performed to verify high order accuracy, well-balanced property, and good resolution for smooth and discontinuous solutions.

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

“…The path integral approach is used to construct a wave propagation algorithm which is well-balanced for isentropic solutions in [8] and isothermal solutions in [9]. A well-balanced weighted essentially nonoscillatory (WENO) finite volume scheme which preserves isothermal hydrostatic solutions is presented in [15] by rewriting the gravitational source terms in an equivalent form that achieves precise balancing. In [6] a second order well-balanced scheme is presented using hydrostatic reconstruction and under the assumption of an isentropic flow for general equation of state; for an ideal gas the isentropic assumption leads to the relation p = Kρ γ .…”

“…A gas-kinetic scheme which is well-balanced for isothermal stationary solutions is presented in [10]. Using the source term formulation of [15], a nonstaggered central scheme which is well-balanced for isothermal stationary solutions is developed in [14]. Well-balanced schemes that satisfy an approximation to the hydrostatic equations have been developed using the approach of relaxation schemes [2,1], in which an approximation to the hydrostatic solution is built into the solution of the approximate Riemann solver.…”

“…The same scheme is automatically well-balanced for both isothermal and polytropic hydrostatic solutions, i.e., it exactly (up to machine precision) preserves the exact hydrostatic solution and this property is independent of the type of gravitational potential. The proposed scheme involves a special discretization of the source terms which is similar to [15] and a reconstruction scheme that uses scaled variables, combined with a numerical flux which preserves stationary contact waves. The scaled variables are chosen so that the pressure is constant in case of the hydrostatic solution, which is crucial to achieving well-balanced property.…”

“…The discretization errors in a non-well-balanced scheme can completely mask the small perturbations. Moreover, even a very high order accurate scheme can lead to an inaccurate prediction of small perturbations if the scheme is not well-balanced [15]. A well-balanced scheme is designed so that it maintains the precise balance of pressure and gravitational forces in case of the hydrostatic solution.…”

A discontinuous Galerkin finite element method for swelling model of polymer gels

“…As we all know, discontinuous Galerkin (DG) methods [11][12][13][14] for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: they allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. And the DG finite element method is also a good method to approximate the displacement-pressure model of kinetics and to alleviate numerical oscillations in the stress field.…”