Jiequan Li

A direct Eulerian GRP scheme for compressible fluid flows

A direct Eulerian generalized Riemann problem (GRP) scheme is derived for compressible fluid flows. Riemann invariants are introduced as the main ingredient to resolve the generalized Riemann problem (GRP) directly for the Eulerian formulation. The crucial auxiliary Lagrangian scheme in the original GRP scheme is not necessary in the present framework. The delicate sonic cases can be easily treated and the extension to multidimensional cases is obtained using the dimensional splitting technique.

Note on the compressible euler equations with zero temperature

Abstract This note presents the behavior of solution of the compressible Euler equations as the temperature drops to zero by the simple riemann problem.

On the Two-Dimensional Gas Expansion for Compressible Euler Equations

We investigate the problem of two-dimensional, unsteady expansion of an inviscid, polytropic gas, which can be interpreted as the collapse of a wedge-shaped dam containing water initially with a uniform velocity. We model this problem by isentropic Euler equations. The flow is quasi-stationary, and using hodograph transform, we describe it by a partial differential equation of second order in the state space if it is irrotational initially. Furthermore, this equation is reduced to a linearly degenerate system of three partial differential equations with inhomogeneous source terms. These properties are used to prove that the flow is globally smooth when a wedge of gas expands into a vacuum, and to analyze that shocks may appear in the interaction of four planar rarefaction waves.

Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier-Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier-Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.

Remark on the generalized Riemann problem method for compressible fluid flows

The generalized Riemann problem (GRP) method was proposed for compressible fluid flows based on the Lagrangian formulation [M. Ben-Artzi, J. Falcovitz, A second-order Godunov-type scheme for compressible fluid dynamics, J. Comput. Phys., 55(1) (1984) 1-32], and a direct Eulerian version was developed in [M. Ben-Artzi, J. Li, G. Warnecke, A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys., 28 (2006) 19-43] by using the concept of Riemann invariants. The central feature of the GRP method is the resolution of centered rarefaction waves. In this note we show how to use the concept of Riemann invariants in order to resolve the rarefaction waves in the Lagrangian coordinate system and result in the GRP scheme.

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.

On the initial-value problem for zero-pressure gas dynamics

This paper is a survey of some recent development on the zero-pressure gas dynamics. We will state the mechanism of delta-shock and its propagation, construct the solutions of one-dimensional and two-dimensional Riemann problems, and then prove the existence of solutions to the general Cauchy problem.

Implementation of the GRP scheme for computing radially symmetric compressible fluid flows

The study of radially symmetric compressible fluid flows is interesting both from the theoretical and numerical points of view. Spherical explosion and implosion in air, water and other media are well-known problems in application. Typical difficulties lie in the treatment of singularity in the geometrical source and the imposition of boundary conditions at the symmetric center, in addition to the resolution of classical discontinuities (shocks and contact discontinuities). In the present paper we present the implementation of direct generalized Riemann problem (GRP) scheme to resolve this issue. The scheme is obtained directly by the time integration of the fluid flows. Our new contribution is to show rigorously that the singularity is removable and derive the updating formulae for mass and energy at the center. Together with the vanishing of the momentum, we obtain new numerical boundary conditions at the center, which are then incorporated into the GRP scheme. The main ingredient is the passage from the Cartesian coordinates to the radially symmetric coordinates.

LOCAL OSCILLATIONS IN FINITE DIFFERENCE SOLUTIONS OF HYPERBOLIC CONSERVATION LAWS

It was generally expected that monotone schemes are oscillation-free for hyperbolic conservation laws. However, recently local oscillations were observed and usually understood to be caused by relative phase errors. In order to further explain this, we first investigate the discretization of initial data that trigger the chequerboard mode, the highest frequency mode. Then we proceed to use the discrete Fourier analysis and the modified equation analysis to distinguish the dissipative and dispersive effects of numerical schemes for low frequency and high frequency modes, respectively. It is shown that the relative phase error is of order ?(1) for the high frequency modes u n j = λ n k iξj , ξ ≈ π, but of order ?(ξ 2 ) for low frequency modes (ξ ≈ 0). In order to avoid numerical oscillations, the relative phase errors should be offset by numerical dissipation of at least the same order. Numerical damping, i.e. the zero order term in the corresponding modified equation, is important to dissipate the oscillations caused by the relative phase errors of high frequency modes. This is in contrast to the role of numerical viscosity, the second order term, which is the lowest order term usually present to suppress the relative phase errors of low frequency modes.

Explicit construction of measure solutions of Cauchy problem for transportation equations

The transportation equations are a mathematical model of zero-pressure flow in gas dynamics and the adhesion particle dynamics system to explain the formation of large scale structures in the universe. With the help of convex hull of a potential function, the solution is explicitly constructed here. It is straightforward to prove that the solution is a global measure one. And Dirac delta-shocks explained as the concentration of particles may develop in the solution.