84 research outputs found
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
Expansion shock waves in regularised shallow water theory
We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularised shallow water equations that include the Benjamin-Bona-Mahoney (BBM) and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock’s existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock
Capturing nonclassical shocks in nonlinear elastodynamic with a conservative finite volume scheme
For a model of nonlinear elastodynamics, we construct a finite volume scheme
which is able to capture nonclassical shocks (also called undercompressive
shocks). Those shocks verify an entropy inequality but are not admissible in
the sense of Liu. They verify a kinetic relation which describes the jump, and
keeps an information on the equilibrium between a vanishing dispersion and a
vanishing diffusion. The scheme pre-sented here is by construction exact when
the initial data is an isolated nonclassical shock. In general, it does not
introduce any diffusion near shocks, and hence nonclas-sical solutions are
correctly approximated. The method is fully conservative and does not use any
shock-tracking mesh. This approach is tested and validated on several test
cases. In particular, as the nonclassical shocks are not diffused at all, it is
possible to obtain large time asymptotics
Stationary expansion shocks for a regularized Boussinesq system
Stationary expansion shocks have been recently identified as a new type of solution to hyperbolic conservation laws regularized by non-local dispersive terms that naturally arise in shallow-water theory. These expansion shocks were studied in [1] for the Benjamin-Bona-Mahony equation using matched asymptotic expansions. In this paper, we extend the analysis of [1] to the regularized Boussinesq system by using Riemann invariants of the underlying dispersionless shallow water equations. The extension for a system is non-trivial, requiring a combination of small amplitude, long-wave expansions with high order matched asymptotics. The constructed asymptotic solution is shown to be in excellent agreement with
accurate numerical simulations of the Boussinesq system for a range of appropriately smoothed Riemann data
Convergent and conservative schemes for nonclassical solutions based on kinetic relations
We propose a new numerical approach to compute nonclassical solutions to
hyperbolic conservation laws. The class of finite difference schemes presented
here is fully conservative and keep nonclassical shock waves as sharp
interfaces, contrary to standard finite difference schemes. The main challenge
is to achieve, at the discretization level, a consistency property with respect
to a prescribed kinetic relation. The latter is required for the selection of
physically meaningful nonclassical shocks. Our method is based on a
reconstruction technique performed in each computational cell that may contain
a nonclassical shock. To validate this approach, we establish several
consistency and stability properties, and we perform careful numerical
experiments. The convergence of the algorithm toward the physically meaningful
solutions selected by a kinetic relation is demonstrated numerically for
several test cases, including concave-convex as well as convex-concave
flux-functions.Comment: 31 page
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