Dimensional Riemann Problems Research Articles

Global structure of two dimensional Riemann solutions to the Euler system for isentropic Chaplygin gas

AbstractIn this paper, we study the two dimensional Riemann problem of the Euler system for isentropic Chaplygin gas, and the initial data consist of three pieces constant states divided by an inverted Y‐type curve. The results extended the results in Chen and Qu where the solution of every one dimensional Riemann problem contains no slip plane. We divide the discussion into several cases and for each case, we give the structure of the solution containing slip planes by the method of generalized characteristic analysis.

High Order Semi-implicit WENO Schemes for All-Mach Full Euler System of Gas Dynamics

In this paper, we propose high order semi-implicit schemes for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. The schemes are proven to be asymptotic preserving and asymptotically accurate as the Mach number vanishes. Besides, they can well capture discontinuous solutions in the compressible regime, especially for two dimensional Riemann problems. Numerical tests in one and two space dimensions will illustrate the effectiveness of the proposed schemes.

Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system

We establish the global existence of subsonic solutions to a two dimensional Riemann problem governed by a self-similar pressure gradient system for shock diffraction by a convex cornered wedge. Since the boundary of the subsonic region consists of a transonic shock and a part of a sonic circle, the governing equation becomes a free boundary problem for nonlinear degenerate elliptic equation of second order with a degenerate oblique derivative boundary condition. We also obtain the optimal \begin{document}$ C^{0,1} $\end{document} -regularity of the solutions across the degenerate sonic boundary.

Numerical Methods for Euler Equations with Self-similar and Quasi Self-similar Solutions

Some problems of Euler equations have self-similar solutions which can be solved by more accurate method. The current paper proposes two new numerical methods for Euler equations with self-similar and quasi self-similar solutions respectively, which can use existing difference schemes for conservation laws and do not need to redesign specified schemes. Numerical experiments are implemented on one dimensional shock tube problems, two dimensional Riemann problems, shock reflection from a solid wedge, and shock refraction at a gaseous interface. For self-similar equations, one-dimensional results are almost equal to the exact solutions, and two-dimensional results also exhibit considerable high resolution. For quasi self-similar equations, the method can solve solutions that are not but close to self-similar, i.e. quasi self-similar, and this method can also achieve very high resolution when computing time is long enough. Numerical simulations to self-similar and quasi self-similar Euler equations have important implications on the study of self-similar problems, development of high resolution schemes, even the research for exact solutions of Euler equations.

Two dimensional Riemann problem for a 2 × 2 system of hyperbolic conservation laws involving three constant states

Zhang and Zheng (1990) conjectured on the structure of a solution for a two-dimensional Riemann problem for Euler equation. To resolve this illuminating conjecture, many researchers have studied the simplified 2 × 2 systems. In this paper, 3-pieces Riemann problem for two-dimensional 2 × 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave. We classify twelve topologically distinct solutions and construct analytical and numerical solutions. The computed numerical solutions clearly confirm the constructed analytic solutions.

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

Some shallow water type models describing the vertical profile of the horizontal velocity with several degrees of freedom have been recently proposed. The question addressed in the current work is the hyperbolicity of a shallow water model with two velocities. The model is written in a nonconservative form and the analysis of its eigenstructure shows the possibility that two eigenvalues coincide. A definition of the nonconservative product is given which enables us to analyse the resonance and coalescence of waves. Eventually, we prove the well-posedness of the two dimensional Riemann problem with initial condition constant by half-plane.

Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions

We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for the two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet.

An accurate predictor-corrector HOC solver for the two dimensional Riemann problem of gas dynamics

The work in the present manuscript is concerned with the simulation of twodimensional (2D) Riemann problem of gas dynamics. We extend our recently developed higher order compact (HOC) method from one-dimensional (1D) to 2D solver and simulate the problem on a square geometry with different initial conditions. The method is fourth order accurate in space and second order accurate in time. We then compare our results with the available benchmark results. The comparison shows an excellent agreement of our results with the existing ones in the literature. Being a finite difference solver, it is quite straight-forward and simple.

Numerical solutions to shock reflection and shock interaction problems for the self-similar transonic two-dimensional nonlinear wave systems

We discuss numerical results for two dimensional Riemann problems for the nonlinear wave system. We consider four sectorial Riemann data and present the numerical results for two configurations. The first configuration is that incident shocks create reflected shocks, and the state behind the reflected shocks becomes subsonic. The second configuration is an interaction between two transonic shocks; one is created by an interaction with the sonic circle, and the other is created by the shock reflection. We implement Lax–Liu positive schemes and Strang splitting, and obtain linear correlations of the incident shock strength and the reflected shock strength for the first configuration. Furthermore we present intriguing numerical results on interactions of two transonic shocks for the second configuration.

An Interaction of a Rarefaction Wave and a Transonic Shock for the Self-Similar Two-Dimensional Nonlinear Wave System

We study a two dimensional Riemann problem for the self-similar nonlinear wave system which gives rise to an interaction of a transonic shock and a rarefaction wave. The interesting feature of this problem is that the governing equation changes its type from supersonic in the far field to subsonic near the origin. The subsonic region is then bounded above by the sonic line (degenerate) and below by the transonic shock (free boundary). Furthermore due to the rarefaction wave in the downstream, which interacts with the transonic shock, the problem becomes inhomogeneous and degenerate. We establish the existence result of the global solution to this configuration, and present analysis to understand the solution structure of this problem.

CENTRAL SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS

The semi-discrete central scheme and central upwind scheme use Runge-Kutta (RK) time discretization. We do the Lax-Wendroff (LW) type time discretization for both schemes. We perform numerical experiments for various problems including two dimensional Riemann problems for Burgers' equation and Euler equations. The results show that the LW time discretization is more efficient in CPU time than the RK time discretization while maintaining the same order of accuracy.

Wave Fronts for Hamilton-Jacobi Equations:¶The General Theory for Riemann Solutions in $\calR^n$

The Hamilton-Jacobi equation describes the dynamics of a hypersurface in \(\). This equation is a nonlinear conservation law and thus has discontinuous solutions. The dependent variable is a surface gradient and the discontinuity is a surface cusp. Here we investigate the intersection of cusp hypersurfaces. These intersections define (n-1)-dimensional Riemann problems for the Hamilton-Jacobi equation. We propose the class of Hamilton-Jacobi equations as a natural higher-dimensional generalization of scalar equations which allow a satisfactory theory of higher-dimensional Riemann problems. The fist main result of this paper is a general framwork for the study of higher-dimensional Riemann problems for Hamilton-Jacobi equations. The purpose of the framwork ist to unterstand the structure of Hamilton-Jacobi wave interactions in an explicit and constructive manner. Specialized to two-dimensional Riemann problems (i.e., the intersection of cusp curves on surfaces embedded in \(\)), this framework provides explicit solutions to a number of cases of interest. We are specifically interested in models of deposition and etching, important processes for the manufacture of semiconductor chips.

Solveur de type Godunov pour simuler les écoulements turbulents compressibles

We present here a Godunov type solver which enables us to compute a non conservative hyperbolic system on unstructured meshes. The convective system comes from the standard turbulent compressible K − ε model. An entropy inequality is given first; then the solution of the associated one dimensional Riemann problem is given, using approximate jump conditions to connect states through discontinuities.