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Register a new userA physical-constraints-preserving genuinely multidimensional HLL scheme for the special relativistic hydrodynamics
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Huazhong Tang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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This paper develops the genuinely multidimensional HLL Riemann solver and finite volume scheme for the two-dimensional special relativistic hydrodynamic equations on Cartesian meshes and studies its physical-constraint-preserving (PCP) property. Several numerical results demonstrate the accuracy, the performance and the resolution of the shock waves and the genuinely multi-dimensional wave structures of the proposed PCP scheme.
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This paper develops entropy stable (ES) adaptive moving mesh schemes for the 2D and 3D special relativistic hydrodynamic (RHD) equations. They are built on the ES finite volume approximation of the RHD equations in curvilinear coordinates, the discrete geometric conservation laws, and the mesh adaptation implemented by iteratively solving the Euler-Lagrange equations of the mesh adaption functional in the computational domain with suitably chosen monitor functions. First, a sufficient condition is proved for the two-point entropy conservative (EC) flux, by mimicking the derivation of the continuous entropy identity in curvilinear coordinates and using the discrete geometric conservation laws given by the conservative metrics method. Based on such sufficient condition, the EC fluxes for the RHD equations in curvilinear coordinates are derived and the second-order accurate semi-discrete EC schemes are developed to satisfy the entropy identity for the given convex entropy pair. Next, the semi-discrete ES schemes satisfying the entropy inequality are proposed by adding a suitable dissipation term to the EC scheme and utilizing linear reconstruction with the minmod limiter in the scaled entropy variables in order to suppress the numerical oscillations of the above EC scheme. Then, the semi-discrete ES schemes are integrated in time by using the second-order strong stability preserving explicit Runge-Kutta schemes. Finally, several numerical results show that our 2D and 3D ES adaptive moving mesh schemes effectively capture the localized structures, such as sharp transitions or discontinuities, and are more efficient than their counterparts on uniform mesh.
This paper extends the second-order accurate BGK finite volume schemes for the ultra-relativistic flow simulations [5] to the 1D and 2D special relativistic hydrodynamics with the Synge equation of state. It is shown that such 2D schemes are very time-consuming due to the moment integrals (triple integrals) so that they are no longer practical. In view of this, the simplified BGK (sBGK) schemes are presented by removing some terms in the approximate nonequilibrium distribution at the cell interface for the BGK scheme without loss of accuracy. They are practical because the moment integrals of the approximate distribution can be reduced to the single integrals by some coordinate transformations. The relations between the left and right states of the shock wave, rarefaction wave, and contact discontinuity are also discussed, so that the exact solution of the 1D Riemann problem could be derived and used for the numerical comparisons. Several numerical experiments are conducted to demonstrate that the proposed gas-kinetic schemes are accurate and stable. A comparison of the sBGK schemes with the BGK scheme in one dimension shows that the former performs almost the same as the latter in terms of the accuracy and resolution, but is much more efficiency.
We present the second-order multidimensional Staggered Grid Hydrodynamics Residual Distribution (SGH RD) scheme for Lagrangian hydrodynamics. The SGH RD scheme is based on the staggered finite element discretizations as in [Dobrev et al., SISC, 2012]. However, the advantage of the residual formulation over classical FEM approaches consists in the natural mass matrix diagonalization which allows one to avoid the solution of the linear system with the global sparse mass matrix while retaining the desired order of accuracy. This is achieved by using Bernstein polynomials as finite element shape functions and coupling the space discretization with the deferred correction type timestepping method. Moreover, it can be shown that for the Lagrangian formulation written in non-conservative form, our residual distribution scheme ensures the exact conservation of the mass, momentum and total energy. In this paper we also discuss construction of numerical viscosity approximations for the SGH RD scheme allowing to reduce the dissipation of the numerical solution. Thanks to the generic formulation of the staggered grid residual distribution scheme, it can be directly applied to both single- and multimaterial and multiphase models. Finally, we demonstrate computational results obtained with the proposed residual distribution scheme for several challenging test problems.
This paper develops high-order accurate entropy stable (ES) adaptive moving mesh finite difference schemes for the two- and three-dimensional special relativistic hydrodynamic (RHD) and magnetohydrodynamic (RMHD) equations, which is the high-order accurate extension of [J.M. Duan and H.Z. Tang, Entropy stable adaptive moving mesh schemes for 2D and 3D special relativistic hydrodynamics, J. Comput. Phys., 426(2021), 109949]. The key point is the derivation of the higher-order accurate entropy conservative (EC) and ES finite difference schemes in the curvilinear coordinates by carefully dealing with the discretization of the temporal and spatial metrics and the Jacobian of the coordinate transformation and constructing the high-order EC and ES fluxes with the discrete metrics. The spatial derivatives in the source terms of the symmetrizable RMHD equations and the geometric conservation laws are discretized by using the linear combinations of the corresponding second-order case to obtain high-order accuracy. Based on the proposed high-order accurate EC schemes and the high-order accurate dissipation terms built on the WENO reconstruction, the high-order accurate ES schemes are obtained for the RHD and RMHD equations in the curvilinear coordinates. The mesh iteration redistribution or adaptive moving mesh strategy is built on the minimization of the mesh adaption functional. Several numerical tests are conducted to validate the shock-capturing ability and high efficiency of our high-order accurate ES adaptive moving mesh methods on the parallel computer system with the MPI communication. The numerical results show that the high-order accurate ES adaptive moving mesh schemes outperform both their counterparts on the uniform mesh and the second-order ES adaptive moving mesh schemes.
This paper explores Tadmors minimum entropy principle for the relativistic hydrodynamics (RHD) equations and incorporates this principle into the design of robust high-order discontinuous Galerkin (DG) and finite volume schemes for RHD on general meshes. The schemes are proven to preserve numerical solutions in a global invariant region constituted by all the known intrinsic constraints: minimum entropy principle, the subluminal constraint on fluid velocity, and the positivity of pressure and rest-mass density. Relativistic effects lead to some essential difficulties in the present study, which are not encountered in the non-relativistic case. Most notably, in the RHD case the specific entropy is a highly nonlinear implicit function of the conservative variables, and, moreover, there is also no explicit formula of the flux in terms of the conservative variables. In order to overcome the resulting challenges, we first propose a novel equivalent form of the invariant region, by skillfully introducing two auxiliary variables. As a notable feature, all the constraints in the novel form are explicit and linear with respect to the conservative variables. This provides a highly effective approach to theoretically analyze the invariant-region-preserving (IRP) property of schemes for RHD, without any assumption on the IRP property of the exact Riemann solver. Based on this, we prove the convexity of the invariant region and establish the generalized Lax--Friedrichs splitting properties via technical estimates, lying the foundation for our IRP analysis. It is shown that the first-order Lax--Friedrichs scheme for RHD satisfies a local minimum entropy principle and is IRP under a CFL condition. Provably IRP high-order DG and finite volume methods are developed for the RHD with the help of a simple scaling limiter. Several numerical examples demonstrate the effectiveness of the proposed schemes.
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