The Riemann problem is solved for 2 x 2 systems of hyperbolic conservation laws having quadratic flux functions. Equations with quadratic flux functions arise from neglecting higher order nonlinear terms in hyperbolic systems that fail to be strictly hyperbolic everywhere. Such equations divide into four classes, three of which are considered in this paper. The solution of the Riemann problem i...

In this paper, we consider a certain class of discrete pseudo-differential operators in a sharp convex cone and describe their invertibility conditions in L2-spaces. For this purpose we introduce a concept of periodic wave factorization for elliptic symbol and show its applicability for the studying. Key–Words: Discrete operator, Multidimensional periodic Riemann problem, Periodic wave factoriz...

We derive a lubrication model describing gravity-driven thin film flow of a suspension of heavy particles in viscous fluid. The main features of this continuum model are an effective mixture viscosity and a particle settling velocity, both depending on particle concentration. The resulting equations form a 2 × 2 system of conservation laws in the film thickness h(x, t) and in φh, where φ(x, t) ...

This paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the Riemann solution for the Euler system of gas dynamics with the co-volume equation of state. The novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy. The convergence rate of the algorithm is cubic a...

We consider front tracking approximate solutions to the p-system of isentropic gas dynamics. At interaction times, the outgoing wave fronts have the same strength as in the exact solution of the Riemann problem, but some error is allowed in their speed. For large BV initial data, we construct examples showing that the total variation of these approximate solutions can become arbitrarily large, ...

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 tre...

Elementary waves in Suliciu model for dynamic phase transitions are obtained through traveling wave analysis. For any given initial data with two pieces of constant states, the Riemann solutions are constructed as a combination of elementary waves. When the initial profile contains three pieces of constant states, the solution may be constructed from the Riemann solutions, with each two adjacen...

We investigate viscous shock profiles of the Riemann problem for systems of hyperbolic balance laws. Even strictly hyperbolic flux terms together with a nonoscillating kinetic part can lead to oscillating viscous shock profiles. They appear near a Hopf-like bifurcation point of the traveling wave equation.

We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set M(r0, s0) = {r0 < r < s < s0}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation VarX(φ(τ(t,X), u(t,X))) is nonincreasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking s...

The one–dimensional differential equations for the conservation mixture mass and mixture momentum coupled by a relative velocity conservation equation have been solved for an isentropic mixture of gas and liquid two–phase flow. Under certain restrictions the resulting system of partial differential equations is hyperbolic and allow discontinuous solutions. These equations have been solved by th...