Wave interactions and stability of the Riemann solutions for a scalar conservation law with a discontinuous flux function

This paper is devoted to studying the interactions of elementary waves for a model of a scalar conservation law with a flux function involving discontinuous coefficients. In order to cover all the situations completely, we take the initial data as three piecewise constant states and the middle region is regarded as the perturbed region with small distance. It is proved that the Riemann solutions are stable under the local small perturbations of the Riemann initial data by letting the perturbed parameter tend to zero. The proof is based on the detailed analysis of the interactions of stationary wave discontinuities with shock waves and rarefaction waves. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

On the approximation of the solutions of the Riemann problem for a discontinuous conservation law

For a class of discontinuous flux functions introduced in [3] (cf. also [4]), we prove, for the Riemann problem, an extension of the existence result proved in [2] for a Lipschitz continuous flux function. In the last section, and based in the previous results, we apply the Lax-Friedrichs approximation method and the limiters technique (cf.[6]) to compute the quoted solution in a numerical example. For related results see [5].     

On decay of periodic entropy solutions to a scalar conservation law

We establish a necessary and sufficient condition for decay of periodic entropy solutions to a multidimensional conservation law with merely continuous flux vector.     

Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law

In this paper, we study the problem of asymptotic stabilization by closed loop feedback for a scalar conservation law with a convex flux and in the context of entropy solutions. Besides the boundary data, we use an additional control which is a source term acting uniformly in space.     

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Centrifugal sedimentation of an ideal suspension in a rotating tube or basket can be modelled by an initial-boundary-value problem for a scalar conservation law with a nonconvex flux function. The sought unknown is the volume fraction of solids as function of radial distance and time for constant initial data. The method of characteristics is used to construct entropy solutions. The qualitatively different solutions, which depend on the initial value and the vessel radial coordinates, are presented in detail along with numerical simulations. Based on the entropy solutions, a new method of flux identification, which does not require any prescribed functional expression, is presented and illustrated with synthetic data.     

Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws

Recently, there has been some interest on the stability of waves where the functions involved grow or decay at an algebraic rate m|x|. In this paper we define the so-called algebraic dichotomy that may aid in treating such problems. We discuss the basic properties of the algebraic dichotomy, methods of detecting it, and calculating the power of the weight function.We present several examples: (1) The Bessel equation. (2) The n-degree Fisher type equation. (3) Hyperbolic conservation laws in similarity coordinates. (4) A system of conservation laws with a Dafermos type viscous regularization. We show that the linearized system generates an analytic semigroup in the space of algebraic decay functions. This example motivates our work on algebraic dichotomies.     

Numerical scheme for a scalar conservation law with memory

We consider approximation of solutions to conservation laws with memory represented by a Volterra term with a smooth decreasing but possibly unbounded kernel. The numerical scheme combines Godunov method with a treatment of the integral term following from product integration rules. We prove stability for both linear and nonlinear flux functions and demonstrate the expected order of convergence using numerical experiments. The problem is motivated by modeling advective transport in heterogeneous media with subscale diffusion.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 239–264, 2014     

A new entropy functional for a scalar conservation law

In this paper we introduce a new entropy functional for a scalar convex conservation law that generalizes the traditional concept of entropy of the second law of thermodynamics. The generalization has two aspects: The new entropy functional is defined not for one but for two solutions. It is defined in terms of the L₁ distance between the two solutions as well as the variations of each separate solution. In addition, it is decreasing in time even when the solutions contain no shocks and is therefore stronger than the traditional entropy even in the case when one of the solutions is zero. © 1999 John Wiley & Sons, Inc.     

Probabilistic viscosity algorithm for the scalar conservation law

We derive an algorithm for solving the initial value problem for u_t = ½σ²u_xx + f(u)u_x. The approach is based on the representation of the solution to the above equation in the form of the functional of Brownian motion. For small σ we get the approximation for u_t = f(u)u_x. A comparison with the random choice method is illustrated by the numerical example.     

Regularity of solutions to scalar conservation laws with a force

We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.     

Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

Well-posedness for a scalar conservation law with singular nonconservative source

We consider the Cauchy problem for the 2×2 strictly hyperbolic system

     

Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws

This paper contains a proof of the existence and uniqueness of solutions to the Riemann problem for systems of two hyperbolic conservation laws in one space variable. Our main assumptions are that the system is strictly hyperbolic and genuinely nonlinear. We also require that the system satisfy standard conditions on the second Fréchet derivatives, and one other hypothesis, which we have called the half-plane condition. This hypothesis replaces other, more restrictive hypotheses required by previous authors. The methods and results of this paper are designed to be applicable to systems of conservation laws which are not strictly hyperbolic.     

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in of the space of directions ζ=x/t. That is, the solution z(t,x,ξ) of the perturbed Cauchy problem for the corresponding BGK system satisfies as t→∞, in , where R(ζ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.     

Viscosity methods for piecewise smooth solutions to scalar conservation laws

It is proved that for scalar conservation laws, if the flux function is strictly convex, and if the entropy solution is piecewise smooth with finitely many discontinuities (which includes initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time and interactions of all these patterns), then the error of viscosity solution to the inviscid solution is bounded by in the -norm, which is an improvement of the upper bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to .