Delta Shock Wave Research Articles

The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws

In this paper, we solve the Riemann problem with the initial data containing Dirac delta functions for a class of coupled hyperbolic systems of conservation laws. Under suitably generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of solutions involving delta shock waves are proved. Further, four kinds of different structure for solutions are established uniquely.

Delta-shock wave type solution of hyperbolic systems of conservation laws

For two classes of hyperbolic systems of conservation laws new definitions of a δ \delta -shock wave type solution are introduced. These two definitions give natural generalizations of the classical definition of the weak solutions. It is relevant to the notion of δ \delta -shocks. The weak asymptotics method developed by the authors is used to describe the propagation of δ \delta -shock waves to the three types of systems of conservation laws and derive the corresponding Rankine–Hugoniot conditions for δ \delta -shocks.

Delta and singular delta locus for one‐dimensional systems of conservation laws

AbstractThis work gives a condition for existence of singular and delta shock wave solutions to Riemann problem for 2×2 systems of conservation laws. For a fixed left‐hand side value of Riemann data, the condition obtained in the paper describes a set of possible right‐hand side values. The procedure is similar to the standard one of finding the Hugoniot locus. Fluxes of the considered systems are globally Lipschitz with respect to one of the dependent variables. The association in a Colombeau‐type algebra is used as a solution concept. Copyright © 2004 John Wiley &Sons, Ltd.

Nonlinear algebraic analysis of delta shock wave solutions to Burgers’ equation

By means of three fundamental structures we can define, in a general way, a sheaf A of differential algebras containing most of the special cases met in the theory of generalized functions. A convenient choice of these structures permits us to study Burgers' equation with δ-Dirac measure as initial data, and we can construct a generalized δ-shock wave as an approximate solution, self-similar to the initial data.

Generalized Plane Delta-Shock Waves for n-Dimensional Zero-Pressure Gas Dynamics

With the help of a generalized plane wave solution, we study a type of generalized plane delta-shock wave for the n-dimensional zero-pressure gas dynamics and refine its generalized Rankine–Hugoniot relation which is a system of ordinary equations. This relation describes accurately the character of the generalized plane delta-shock: location, propagation speed, and weight. Under a suitable entropy condition, four different explicit constructions of solutions for a kind of Riemann problem with Radon measure as initial data are established uniquely. The overtaking of two plane delta-shocks forming a new generalized plane delta-shock is also investigated. Finally, the 2-D Riemann problem with four pieces of initial data is solved in a simplified situation.

Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics

In this paper we study zero-pressure gas dynamics, which is a nonstrict hyperbolic system of nonlinear conservation laws with delta-shock waves in solutions. By using the generalized Rankine-Hugoniot relations to solve the Riemann problem with two pieces of constant initial data, multidimensional planar delta-shock waves dependent upon a family of one parameter are obtained. Furthermore, we choose a unique entropy solution through the process of a viscosity vanishing, and obtain a stability for delta-shocks in multidimensions.

Delta-shock waves as self-similar viscosity limits

Delta-shock waves as self-similar viscosity limits

On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities

We are concerned with the Riemann problem for the two-dimensional compressible Euler equations in gas dynamics. This paper is a continuation of our program (see [CY1,CY2]) in studying the interaction of nonlinear waves in the Riemann problem. The central point in this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that connect two neighboring constant initial states in the quadrants. In this paper we focus mainly on the interaction of contact discontinuities, which consists of two genuinely different cases. For each case, the structure of the Riemann solution is analyzed by using the method of characteristics, and the corresponding numerical solution is illustrated via contour plots by using the upwind averaging scheme that is second-order in the smooth region of the solution developed in [CY1]. For one case, the four contact discontinuities role up and generate a vortex, and the density monotonically decreases to zero at the center of the vortex along the stream curves. For the other, two shock waves are formed and, in the subsonic region between two shock waves, a new kind of nonlinear hyperbolic waves (called smoothed Delta-shock waves) is observed.

Riemann Problems for a Class of Coupled Hyperbolic Systems of Conservation Laws

We study a class of coupled hyperbolic systems of conservation laws which contain the one-dimensional zero-pressure gas dynamics as a prototypical example. The Riemann problems are solved constructively. The Riemann solutions exactly include two kinds: delta-shock wave solutions and vacuum solutions. Under the generalized Rankine–Hugoniot relation and entropy condition, all of the existence, uniqueness, and stability of solutions to viscous perturbations are proved. Two typical examples are presented finally.

A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws

A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws

Delta-Shock Waves as Limits of Vanishing Viscosity for Hyperbolic Systems of Conservation Laws

For simple models of hyperbolic systems of conservation laws, we study a new type of nonlinear hyperbolic wave, a delta-shock wave, which is a Dirac delta function supported on a shock. We prove that delta-shock waves are w*-limits in L 1 of solutions to some reasonable viscous perturbations as the viscosity vanishes. Further, we solve the multiplication problem of a delta function with a discontinuous function to show that delta-shock waves satisfy the equations in the sense of distributions. Under suitable generalized Rankine-Hugoniot and entropy conditions, we establish the existence and uniqueness of solutions involving delta-shock waves for the Riemann problems. The existence of solutions to the Cauchy problem is also investigated.