Abstract. Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form f mare any consistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.
Solving the Euler equations of ideal hydrodynamics as accurately and efficiently as possible is a key requirement in many astrophysical simulations. It is therefore important to continuously advance the numerical methods implemented in current astrophysical codes, especially also in light of evolving computer technology, which favours certain computational approaches over others. Here we introduce the new adaptive mesh refinement (AMR) code TENET, which employs a high order discontinuous Galerkin (DG) scheme for hydrodynamics. The Euler equations in this method are solved in a weak formulation with a polynomial basis by means of explicit Runge-Kutta time integration and Gauss-Legendre quadrature. This approach offers significant advantages over commonly employed second order finite volume (FV) solvers. In particular, the higher order capability renders it computationally more efficient, in the sense that the same precision can be obtained at significantly less computational cost. Also, the DG scheme inherently conserves angular momentum in regions where no limiting takes place, and it typically produces much smaller numerical diffusion and advection errors than a FV approach. A further advantage lies in a more natural handling of AMR refinement boundaries, where a fall-back to first order can be avoided. Finally, DG requires no wide stencils at high order, and offers an improved data locality and a focus on local computations, which is favourable for current and upcoming highly parallel supercomputers. We describe the formulation and implementation details of our new code, and demonstrate its performance and accuracy with a set of two-and three-dimensional test problems. The results confirm that DG schemes have a high potential for astrophysical applications.
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. Introduction.Conservation laws with gravitational source terms occur in many PDE models like shallow water equations and Euler equations. These equations possess nontrivial stationary solutions which are referred to as hydrostatic solutions in the case of Euler equations. Euler equations with gravity are useful models for atmospheric flows and stellar structure simulations in astrophysical applications. The hydrostatic Euler equation takes the form of an ordinary differential equation in which the pressure forces are balanced by the gravitational forces. This precise balancing has to be achieved at the numerical level in order to preserve the stationary solution. Since the gravitational source terms are nonconservative, this precise balancing is not easy to achieve in the numerical scheme. Conventional numerical schemes in which the source term may be discretized in a consistent manner are not able to preserve such stationary solutions especially on coarse meshes. This leads to erroneous numerical solutions especially when trying to compute small perturbations around the stationary solution necessitating the need for very fine meshes. However, in practical threedimensional (3-D) computations it may not be possible to use very fine meshes. 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. This enables such schemes to more accurately resolve small perturbations around the stationary solution.To solve the hydrostatic Euler equations exactly, we have to make additional assumptions, for example, constant temperature or constant entropy or a more general
We present a finite volume scheme for ideal compressible magnetohydrodynamic (MHD) equations on 2-D Cartesian meshes. The semi-discrete scheme is constructed to be entropy stable by using the symmetrized version of the equations as introduced by Godunov. We first construct an entropy conservative scheme for which su cient condition is given and we also derive a numerical flux satisfying this condition. Secondly, following a standard procedure, we make the scheme entropy stable by adding dissipative flux terms using jumps in entropy variables. A semidiscrete high resolution scheme is constructed that preserves the entropy stability of the first order scheme. We demonstrate the robustness of this new scheme on several standard MHD test cases.
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.
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