Huazhong Tang

Kailiang Wu, Huazhong Tang, Dongbin Xiu

This paper is concerned with generalized polynomial chaos (gPC) approximation for a general system of quasilinear hyperbolic conservation laws with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Stochastic Galerkin method is then applied to derive the equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is proved to be symmetrically hyperbolic. This important property then allows one to use a variety of numerical schemes for spatial and temporal discretization. Here a higher-order and path-conservative finite volume WENO scheme is adopted in space, along with a third-order total variation diminishing Runge-Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional system is carried over via the operator splitting technique. Several 1D and 2D numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

Kailiang Wu, Huazhong Tang

This paper first studies the admissible state set $\mathcal G$ of relativistic magnetohydrodynamics (RMHD). It paves a way for developing physical-constraints-preserving (PCP) schemes for RMHD equations with the solutions in $\mathcal G$. To overcome the difficulties arising from the extremely strong nonlinearities and no explicit formulas of the primitive variables and flux vectors with respect to the conservative vector, two equivalent forms of $\mathcal G$ with explicit constraints on the conservative vector are skillfully discovered. The first is derived by analyzing roots of several polynomials and transferring successively them, and further used to prove the convexity of $\mathcal G$ with the aid of semi-positive definiteness of the second fundamental form of a hypersurface. While the second is derived based on the convexity and then used to show the orthogonal invariance of $\mathcal G$. The Lax-Friedrichs (LxF) splitting property does not hold generally for nonzero magnetic field, but by a constructive inequality and pivotal techniques, we discover the generalized LxF splitting properties, combining the convex combination of some LxF splitting terms with a discrete divergence-free condition of magnetic field. Based on above analyses, several one- and two-dimensional PCP schemes are then studied. In 1D case, a first-order accurate LxF type scheme is first proved to be PCP under the CFL condition, and then high-order PCP schemes are proposed via a PCP limiter. In 2D case, the discrete divergence-free condition and PCP property are analyzed for a first-order LxF scheme, and two sufficient conditions are derived for high-order accurate PCP schemes. Our analysis reveals in theory for the first time that the discrete divergence-free condition is closely connected with the PCP property. Several numerical examples demonstrate theoretical findings and performance of numerical schemes.

Jian Zhao, Huazhong Tang

This paper develops $P^K$-based non-central and central Runge-Kutta discontinuous Galerkin (DG) methods with WENO limiter for the one- and two-dimensional special relativistic magnetohydrodynamical (RMHD) equations, $K=1,2,3$. The non-central DG methods are locally divergence-free, while the central DG are "exactly" divergence-free but have to find two approximate solutions defined on mutually dual meshes. The adaptive WENO limiter first identifies the "troubled" cells by using a modified TVB minmod function, and then uses the WENO technique to locally reconstruct a new polynomial of degree $(2K+1)$ inside the "troubled" cells replacing the DG solution by based on the cell average values of the DG solutions in the neighboring cells as well as the original cell averages of the "troubled" cells. The WENO limiting procedure does not destroy the locally or "exactly" divergence-free property of magnetic field and is only employed for finite "troubled" cells so that the computational cost can be as little as possible. Several test problems in one and two dimensions are solved by using our non-central and central Runge-Kutta DG methods with WENO limiter. The numerical results demonstrate that our methods are stable, accurate, and robust in resolving complex wave structures.

Kailiang Wu, Huazhong Tang

The paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the spherically symmetric general relativistic hydrodynamical (RHD) equations and a second-order accurate discretization for the spherically symmetric Einstein (SSE) equations. The former is directly using the Riemann invariants and the Runkine-Hugoniot jump conditions to analytically resolve the left and right nonlinear waves of the local GRP in the Eulerian formulation together with the local change of the metrics to obtain the limiting values of the time derivatives of the conservative variables along the cell interface and the numerical flux for the GRP scheme. While the latter utilizes the energy-momentum tensor obtained in the GRP solver to evaluate the fluid variables in the SSE equations and keeps the continuity of the metrics at the cell interfaces. Several numerical experiments show that the GRP scheme can achieve second-order accuracy and high resolution, and is effective for spherically symmetric general RHD problems.

Kailiang Wu, Huazhong Tang

The paper studies the physical-constraints-preserving (PCP) schemes for multi-dimensional special relativistic magnetohydrodynamics with a general equation of state (EOS) on more general meshes. It is an extension of the work [Math. Models Methods Appl. Sci., 27:1871-1928, 2017] which focuses on the ideal EOS and uniform Cartesian meshes. The general EOS without a special expression poses some additional difficulties in discussing the mathematical properties of admissible state set with the physical constraints on the fluid velocity, density and pressure. Rigorous analyses are provided for the PCP property of finite volume or discontinuous Galerkin schemes with the Lax-Friedrichs (LxF) type flux on a general mesh with non-self-intersecting polytopes. Those are built on a more general form of generalized LxF splitting property and a different convex decomposition technique. It is shown in theory that the PCP property is closely connected with a discrete divergence-free condition, which is proposed on the general mesh and milder than that in [Math. Models Methods Appl. Sci., 27:1871-1928, 2017].

Yuhuan Yuan, Huazhong Tang

This paper studies the two-stage fourth-order accurate time discretization \cite{LI-DU:2016} and applies it to special relativistic hydrodynamical equations. It is shown that new two-stage fourth-order accurate time discretizations can be proposed. With the aid of the direct Eulerian GRP (generalized Riemann problem) methods \cite{Yang-He-Tang:2011,Yang-Tang:2012} and the analytical resolution of the local "quasi 1D" GRP, the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations. Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.

Yaping Chen, Yangyu Kuang, Huazhong Tang

This paper presents second-order accurate genuine BGK (Bhatnagar-Gross-Krook) schemes in the framework of finite volume method for the ultra-relativistic flows. Different from the existing kinetic flux-vector splitting (KFVS) or BGK-type schemes for the ultra-relativistic Euler equations, the present genuine BGK schemes are derived from the analytical solution of the Anderson-Witting model, which is given for the first time and includes the "genuine" particle collisions in the gas transport process. The BGK schemes for the ultra-relativistic viscous flows are also developed and two examples of ultra-relativistic viscous flow are designed. Several 1D and 2D numerical experiments are conducted to demonstrate that the proposed BGK schemes not only are accurate and stable in simulating ultra-relativistic inviscid and viscous flows, but also have higher resolution at the contact discontinuity than the KFVS or BGK-type schemes.

Yangyu Kuang, Huazhong Tang

This paper extends the model reduction method by the operator projection to the three-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order moment system is built on our careful study of infinite families of the complicate Grad type orthogonal polynomials depending on a parameter and the real spherical harmonics. We derive the recurrence relations of the polynomials, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros. The recurrence relations and partial derivatives of the real spherical harmonics are also given. It is proved that our moment system is globally hyperbolic, and linearly stable. Moreover, the Lorentz covariance is also studied in the 1D space.

Yangyu Kuang, Huazhong Tang

This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of two families of the complicate Grad type orthogonal polynomials depending on a parameter. We derive their recurrence relations, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros and coefficient matrices in the recurrence formulas. Some properties of the moment system are also proved. They include the eigenvalues and their bound as well as eigenvectors,hyperbolicity, characteristic fields, linear stability, and Lorentz covariance. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. The results show that the solutions of our hyperbolic moment system can converge to the solution of the special relativistic Boltzmann equation as the order of the hyperbolic moment system increases.

Junming Duan, Yangyu Kuang, Huazhong Tang

This paper derives the arbitrary order globally hyperbolic moment system for a non-linear kinetic description of the Vicsek swarming model by using the operator projection. It is built on our careful study of a family of the complicate Grad type orthogonal functions depending on a parameter (angle of macroscopic velocity). We calculate their derivatives with respect to the independent vari- able, and projection of those derivatives, the product of velocity and basis, and collision term. The moment system is also proved to be hyperbolic, rotational invariant, and mass-conservative. The relationship between Grad type expansions in different parameter is also established. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. It is also compared to the spectral method for the kinetic equation. The results show that the solutions of our hyperbolic moment system converge to the solutions of the kinetic equation for the Vicsek model as the order of the moment system increases, and the moment method can capture key features such as rarefaction and shock waves, contact discontinuity, and vortex formation.

Yangyu Kuang, Huazhong Tang

The paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the radiation hydrodynamical equations (RHE) in the zero diffusion limit. The difficulty comes from no explicit expression of the flux in terms of the conservative vector. The characteristic fields and the relations between the left and right states across the elementary-waves are first studied, and then the solution of the one-dimensional Riemann problem is analyzed and given. Based on those, the direct Eulerian GRP scheme is derived by directly using the generalized Riemann invariants and the Runkine-Hugoniot jump conditions to analytically resolve the left and right nonlinear waves of the local GRP in the Eulerian formulation. Several numerical examples show that the GRP scheme can achieve second-order accuracy and high resolution of strong discontinuity.

Junming Duan, Huazhong Tang

This paper develops the high-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semi-discrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order schemes. Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.

Jian Zhao, Huazhong Tang

This paper developes Runge-Kutta $P^K$-based central discontinuous Galerkin (CDG) methods with WENO limiter to the one- and two-dimensional special relativistic hydrodynamical (RHD) equations, $K=1,2,3$. Different from the non-central DG methods, the \CDG{} have to find two approximate solutions defined on mutually dual meshes. For each mesh, the CDG approximate solutions on its dual mesh are used to calculate the flux values in the cell and on the cell boundary so that the approximate solutions on mutually dual meshes are coupled with each other, and the use of numerical flux may be avoided. The WENO limiter is adaptively implemented via two steps: the "troubled" cells are first identified by using a modified TVB minmod function, and then the WENO technique is used to locally reconstruct new polynomials of degree $(2K+1)$ replacing the CDG solutions inside the "troubled' cells by the cell average values of the CDG solutions in the neighboring cells as well as the original cell averages of the "troubled" cells. Because the WENO limiter is only employed for finite "troubled" cells, the computational cost can be as little as possible. The accuracy of the CDG without the numerical dissipation is analyzed and calculation of the flux integrals over the cells is also addressed. Several test problems in one and two dimensions are solved by using our \CDG{} with WENO limiter. The computations demonstrate that our methods are stable, accurate, and robust in solving complex RHD problems.

Yangyu Kuang, Kailiang Wu, Huazhong Tang

The paper develops high order accurate Runge-Kutta discontinuous local evolution Galerkin (RKDLEG) methods on the cubed-sphere grid for the shallow water equations (SWEs). Instead of using the dimensional splitting method or solving one-dimensional Riemann problem in the direction normal to the cell interface, the RKDLEG methods are built on genuinely multi-dimensional approximate local evolution operator of the locally linearized SWEs on a sphere by considering all bicharacteristic directions. Several numerical experiments are conducted to demonstrate the accuracy and performance of our RKDLEG methods, in comparison to the Runge-Kutta discontinuous Galerkin method with Godunov's flux etc.

Kailiang Wu, Huazhong Tang

The paper develops a Newton multigrid (MG) method for one- and two-dimensional steady-state shallow water equations (SWEs) with topography and dry areas.It solves the nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs by using Newton's method as the outer iteration and a geometric MG method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton MG method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transitions. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton MG method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.

Kailiang Wu, Huazhong Tang

The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics. The paper develops high-order accurate physical-constraints-preserving (PCP) central discontinuous Galerkin (DG) methods for the one- and two-dimensional special relativistic hydrodynamic (RHD) equations with a general EOS. It is built on the theoretical analysis of the admissible states for the RHD and the PCP limiting procedure enforcing the admissibility of central DG solutions. The convexity, scaling and orthogonal invariance, and Lax-Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form, and then the high-order central DG methods with the PCP limiting procedure and strong stability preserving time discretization are proved to preserve the positivity of the density, pressure, and specific internal energy, and the bound of the fluid velocity, maintain the high-order accuracy, and be $L^1$-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density or pressure etc.

Kailiang Wu, Huazhong Tang

The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrich splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit expressions of the primitive variables and the flux vectors, in terms of the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector replacing the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case. Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.