Riemann problem

P. Colella and H.M. Glaz, Efficient Solution Algorithms for the Riemann Problem for Real Gasses, J. Comput. Phys. 59 (1985). 

Figure 3. Nontrivial solution of a degenerate Riemann problem with initial data in the metastable area.

Shearer, M., and Y. Y ang, 1993, The Riemann problem for a p-system of conservation laws of mixed type with cubic nonlinearity, Proc.Roy.Soc. Edinburgh A. ... 

To numerically solve Eq. (25), the finite volume method is used. In doing so, a mesh consisting of triangle cells is constructed at the TE surface in each cell, the flux parameters are taken to be constant. The fluxes through the boundaries of finite volumes are determined on the basis of the solution of the onedimensional Riemann problem in the direction of normal to the boundary between cells. 

The conservation law together with piecewise constant data having a single discontinuity is known as the Riemann problem [117]. 

In the next two sections we discuss the Riemann problem where the initial data is given by... 

Hattori, H., The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion - Isothermal case, Arch. Rat. Mech. Anal. 92 (1986), 247-263. 

Slemrod, M. A limiting viscosity approach to the Riemann problem for materials exhibiting change of phase. Archive for Rational Mechanics and Analysis 105 (1989) 327-365. 

The mathematical justification for the assumption of coherent concentration fronts proceeds via the inverse of this argument. It has been shown that, in a system governed by the kinematic wave equation [Eq. (9.12)] involving a transition between two constant states (a Riemann problem), if a unique solution exists it is always a function of the combined variable the... 

Model 2. Another estimate of Vc and P43 is obtained by analyzing a one-dimensional Riemann problem for the case when the SW front interacts with a planar CS and a subsequent direct reflection of a planar SW from a... 

The problem of the linear relationship may be stated as follows given the equation connecting the boundary values of an analytic function on a line of discontinuity together with some condition on the function at infinity, determine the analytic function, either uniquely or very nearly uniquely. This has been termed the Hilbert problem by Muskhelishvili (1953, 1963) and the Riemann problem by Gakhov (1966). We adopt the former terminology here. 

The theory of hyperbolic systems makes it possible to solve the problem known as the Riemann Problem (R.P.) where one seeks to connect two constant states (see also Glueckauf, 1949). It corresponds to the transfirrmation of an homogeneous rock which is the downstream medium (first constant state) by a fluid of constant composition (upstream medium, second constant state). Because the fluid circulates from the left to the right, the downstream medium is said on the right ("a droite", d) and the upstream medium is said on the left ("a gauche", g)). The theory shows that the solution of the R.P. is made of 3 constant states in addition to the two states (g) and (d), another constant state (intermediate, i) appears, which is connected to (g) and (d) by shocks, "detentes" or contact... 

Figure 3. Representation in both the composition space (left) and the (composition, space) coordinates (rig)it) of the evolution of solid solution composition for different types of initial and boundary conditions, giving rise to different types of Riemann problems. The two independent composition variables arc termed u and v. The craistant states of the Riemann problems are called u , v and u v" respectively for each case. Shock, ddtente waves appear, so as intermediate plateaus. The 1-waves and 2-waves are labelled by I and 2 respectively.

State is the common term for the solution of a Riemann problem. 

Fig. 6. The generalized Riemann problem along the outward pointing unit vector with reconstructed polynomials Pin(x) and Pout(x) approximating the solution u.

GRPo denotes the conventional Riemann Problem (RP) with two constant functions as initial condition. 

E.F. Toro and V.A. Titarev (2002) Solution of the generalized Riemann problem for advection-reaction equations. Proc. R. Soc. Loud. A, 271-281. 

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