Huazhong Tang

Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws

We develop efficient moving mesh algorithms for one- and two-dimensional hyperbolic systems of conservation laws. The algorithms are formed by two independent parts: PDE evolution and mesh-redistribution. The first part can be any appropriate high-resolution scheme, and the second part is based on an iterative procedure. In each iteration, meshes are first redistributed by an equidistribution principle, and then on the resulting new grids the underlying numerical solutions are updated by a conservative-interpolation formula proposed in this work. The iteration for the mesh-redistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state. The main idea of the proposed method is to keep the mass-conservation of the underlying numerical solution at each redistribution step. In one dimension, we can show that the underlying numerical approximation obtained in the mesh-redistribution part satisfies the desired TVD property, which guarantees that the numerical solution at any time level is TVD, provided that the PDE solver in the first part satisfies such a property. Several test problems in one and two dimensions are computed using the proposed moving mesh algorithm. The computations demonstrate that our methods are efficient for solving problems with shock discontinuities, obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach.

An adaptive mesh redistribution method for nonlinear Hamilton--Jacobi equations in two-and three-dimensions

This paper presents an adaptive mesh redistribution (AMR) method for solving the nonlinear Hamilton-Jacobi equations and level-set equations in two- and three-dimensions. Our approach includes two key ingredients: a nonconservative second-order interpolation on the updated adaptive grids, and a class of monitor functions (or indicators) suitable for the Hamilton-Jacobi problems. The proposed adaptive mesh methods transform a uniform mesh in the logical domain to cluster grid points at the regions of the physical domain where the solution or its derivative is singular or nearly singular. Moreover, the formal second-order rate of convergence is preserved for the proposed AMR methods. Extensive numerical experiments are performed to demonstrate the efficiency and robustness of the proposed adaptive mesh algorithm.

An adaptive ghost fluid finite volume method for compressible gas-water simulations

An adaptive ghost fluid finite volume method is developed for one- and two-dimensional compressible multi-medium flows in this work. It couples the real ghost fluid method (GFM) [C.W. Wang, T.G. Liu, B.C. Khoo, A real-ghost fluid method for the simulation of multi-medium compressible flow, SIAM J. Sci. Comput. 28 (2006) 278-302] and the adaptive moving mesh method [H.Z. Tang, T. Tang. Moving mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487-515; H.Z. Tang, T. Tang, P.W. Zhang, An adaptive mesh redistribution method for non-linear Hamilton-Jacobi equations in two- and three-dimensions, J. Comput. Phys. 188 (2003) 543-572], and thus combines their advantages. This work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the GFM around the material interface and other discontinuities. Besides the improvement of flow field resolution, the adaptive GFM also largely increases the computational efficiency. Several numerical experiments are conducted to demonstrate robustness and efficiency of the current method. They include several 1D and 2D gas-water flow problems, involving a large density gradient at the material interface and strong shock-interface interactions. The results show that our algorithm can capture the shock waves and the material interface accurately, and is stable and robust even for solutions with large density and pressure gradients.

A note on the conservative schemes for the Euler equations

This note gives a numerical investigation for the popular high resolution conservative schemes when applied to inviscid, compressible, perfect gas flows with an initial high density ratio as well as a high pressure ratio. The results show that they work very inefficiently and may give inaccurate numerical results even over a very fine mesh when applied to such a problem. Numerical tests show that increasing the order of accuracy of the numerical schemes does not help much in improving the numerical results. How to cure this difficulty is still open.

An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics

This paper presents an adaptive moving mesh algorithm for two-dimensional (2D) ideal magnetohydrodynamics (MHD) that utilizes a staggered constrained transport technique to keep the magnetic field divergence-free. The algorithm consists of two independent parts: MHD evolution and mesh-redistribution. The first part is a high-resolution, divergence-free, shock-capturing scheme on a fixed quadrangular mesh, while the second part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservative-interpolation formula is used to calculate the remapped cell-averages of the mass, momentum, and total energy on the resulting new mesh; the magnetic potential is remapped to the new mesh in a non-conservative way and is reconstructed to give a divergence-free magnetic field on the new mesh. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy, track and resolve strong shock waves in ideal MHD problems, and preserve divergence-free property of the magnetic field. Numerical examples include the smooth Alfven wave problem, 2D and 2.5D shock tube problems, two rotor problems, the stringent blast problem, and the cloud-shock interaction problem.

A direct Eulerian GRP scheme for relativistic hydrodynamics: Two-dimensional case

This paper develops a direct Eulerian generalized Riemann problem (GRP) scheme for two-dimensional (2D) relativistic hydrodynamics (RHD). It is an extension of the GRP scheme for one-dimensional (1D) RHDs [Z.C. Yang, P. He, H.Z. Tang, J. Comput. Phys. 230 (2011) 7964-7987] and the GRP scheme for the non-relativistic hydrodynamics [M. Ben-Artzi, J.Q. Li, G. Warnecke, J. Comput. Phys. 218 (2006) 19-43]. In order to derive the direct Eulerian GRP scheme, the (local) GRP of the split 2D RHD equations in the Eulerian formulation has to be directly resolved by using corresponding Riemann invariants and Rankine-Hugoniot jump conditions so that the crucial and delicate Lagrangian treatment in the original GRP scheme [M. Ben-Artzi, J. Falcovitz, J. Comput. Phys. 55 (1984) 1-32] may be avoided. An important difference of resolving the GRP of the split 2D RHD equations from the GRP of the 1D RHD equations or the non-relativistic hydrodynamical equations is coming from the fact that the flow regions across the shock or rarefaction wave in the GRP of the split 2D RHD equations are nonlinearly coupled through the Lorentz factor which is also built in terms of the tangential velocities. It is a purely multi-dimensional relativistic feature. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed 2D GRP scheme.

High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrichs 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 20. 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 formulas of the primitive variables and the flux vectors with respect to 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 instead of 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.

An adaptive GRP scheme for compressible fluid flows

This paper presents a second-order accurate adaptive generalized Riemann problem (GRP) scheme for one and two dimensional compressible fluid flows. The current scheme consists of two independent parts: Mesh redistribution and PDE evolution. The first part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservative interpolation formula is used to calculate the cell-averages and the slopes of conservative variables on the resulting new mesh. The second part is to evolve the compressible fluid flows on a fixed nonuniform mesh with the Eulerian GRP scheme, which is directly extended to two-dimensional arbitrary quadrilateral meshes. Several numerical examples show that the current adaptive GRP scheme does not only improve the resolution as well as accuracy of numerical solutions with a few mesh points, but also reduces possible errors or oscillations effectively.

NUMERICAL SIMULATIONS OF RESONANT OSCILLATIONS IN A TUBE

The article concerns a genuinely two-dimensional numerical study of resonant oscillation phenomena in a gas-filled tube with an isothermal wall or an adiabatic wall. The time-dependent, axisymmetric, compressible Navier-Stokes equations in two dimensions are solved by a new finite volume method with the second-order kinetic flux-vector splitting (KFVS) scheme for convective terms, and a third-order Runge-Kutta method for the time evolution. The oscillatory motion of the fluid in a closed tube is generated by a piston at one end, and reflected by the other closed end. Weak shock waves propagating within the tube at the resonant frequency and slightly off-resonance frequencies are numerically captured, which are consistent with both experimental observation and previous theoretical analyses. The interesting results of the sudden change in axial velocity near the piston and the closed end are also presented.The article concerns a genuinely two-dimensional numerical study of resonant oscillation phenomena in a gas-filled tube with an isothermal wall or an adiabatic wall. The time-dependent, axisymmetric, compressible Navier-Stokes equations in two dimensions are solved by a new finite volume method with the second-order kinetic flux-vector splitting (KFVS) scheme for convective terms, and a third-order Runge-Kutta method for the time evolution. The oscillatory motion of the fluid in a closed tube is generated by a piston at one end, and reflected by the other closed end. Weak shock waves propagating within the tube at the resonant frequency and slightly off-resonance frequencies are numerically captured, which are consistent with both experimental observation and previous theoretical analyses. The interesting results of the sudden change in axial velocity near the piston and the closed end are also presented.

A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics

The paper proposes and implements a third-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for one- and two-dimensional (1D & 2D) Euler equations in gas dynamics. It is an extension of the second-order accurate GRP scheme proposed in Ben-Artzi et al. (2006) [5]. The approximate states in numerical fluxes of the third-order accurate GRP scheme are derived by using the higher-order WENO reconstruction of the initial data, the limiting values of the time derivatives of the solutions at the singularity point, and the Jacobian matrix. Besides the limiting values of the first-order time derivatives of fluid variables, the second-order time derivatives are also needed in developing the present GRP scheme and obtained by directly and analytically resolving the local GRP in the Eulerian formulation via two main ingredients, i.e. the Riemann invariants and Rankine-Hugoniot jump conditions. Unfortunately, for the sonic case that the transonic rarefaction wave appears in the GRP, the Jacobian matrix is singular on the sonic line. To this end, those approximate states are given in a different way that is based on the analytical resolution of the transonic rarefaction wave and the local quadratic polynomial interpolation. The 2D GRP scheme is implemented by using the third-order accurate time-splitting method. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed GRP scheme, in comparison to the second-order accurate GRP scheme.