This paper concerns 2 2 systems of conservation laws with quadratic uxes corresponding to Case II in the classiication of such problems. Aided by a computer program , we have constructed the solution that satisses the viscous proole criterion for shock admissibility. Our solution diiers from that obtained using the Lax admissibility criterion, even though solutions exist and are unique for both...

We develop and analyze an improved version of the Multi-Stage (MUSTA) approach to the construction of upwind Godunov-type fluxes whereby the solution of the Riemann problem, approximate or exact, is not required. The new MUSTA schemes improve upon the original schemes in terms of monotonicity properties, accuracy and stability in multiple space dimensions. We incorporate the MUSTA technology in...

It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2D Riemann problem, defined by the interaction of four rarefaction waves. We consider both the customary configu...

Here, U ≡ (v,ux,uy,e), superscript T stands for transpose, r is mass density, v ≡ 1/ρ, ux and uy are the components of flow velocity, e is the specific total energy, and Fx and Fy are fluxes in the xand ydirections respectively. A 2-D Riemann problem is Eq.(1) with a set of constant states for each region surrounding a point; for example, four constant states in the four quadrants in a structur...

In this paper, we study systems of two conservation laws with homogeneous quadratic flux functions. We use the viscous profile criterion for shock admissibility. This criterion leads to the occurrence of non-classical transitional shock waves, which are sensitively dependent on the form of the viscosity matrix. The goal of this paper is to lay a foundation for investigating how the structure of...

We introduce a hierarchy of integrable partial differential equations in 2 + 1 dimensions arising from the commutation of 2 dimensional vector fields, and we construct the formal solution of the associated Cauchy problems using the Inverse Scattering Transform for one-parameter families of vector fields recently introduced in [1, 2]. Due to the ring property of the space of eigenfunctions, the ...

We provide a precise description of the set of residual boundary conditions generated by the self-similar viscous approximation introduced by Dafermos et al. We then apply our results, valid for both conservative and non conservative systems, to the analysis of the boundary Riemann problem and we show that, under appropriate assumptions, the limits of the self-similar and the classical vanishin...

*Correspondence: [email protected] School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, P.R. China Abstract The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws is considered. By the properties of wave curves, we observe that this system does not contain the composite wave compared to the barotropic models of gas dynamics with differ...

We present here a three-fluid three-pressure model to describe three-field patterns or three-phase flows. The basic ideas rely on the counterpart of the two-fluid two-pressure model which has been introduced in the DDT (deflagration to detonation theory) framework, and has been more recently extended to liquid–vapour simulations. We first show that the system is hyperbolic without any constrain...

We present numerical solutions of a two-dimensional Riemann problem for the unsteady transonic small disturbance equations that provides an asymptotic description of the Mach reflection of weak shock waves. We develop a new numerical scheme to solve the equations in selfsimilar coordinates and use local grid refinement to resolve the solution in the reflection region. The solutions contain a re...