In this paper we study the asymptotic equivalence of a general system of 1-D conservation laws and the corresponding relaxation model proposed by S. Jin and Z. Xin (1995, Comm. Pure Appl. Math. 48, 235 277) in the limit of small relaxation rate. It is shown that if the relaxation system satisfies the subcharacteristic condition and the solution of the hyperbolic conservation laws is piecewise s...

Several recently developed multisymplectic schemes for Hamiltonian PDEs have been shown to preserve associated local conservation laws and constraints very well in long time numerical simulations. Backward error analysis for PDEs, or the method of modified equations, is a useful technique for studying the qualitative behavior of a discretization and provides insight into the preservation proper...

Fourth-order nonlinear diffusion equations appear frequently in the description of physical processes, among these, the lubrication equation or the corresponding modified version play an important role in the study of the interface movements. In this work, we use the free software MAXIMA program symmgrp2009.max derived by W. Heremann to calculate the determining equations for the classical symm...

Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to nonlinear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy soluti...

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract Motivated by issues arising for discrete second-order conservation laws and their continuum limits (applicable, for example, to one-dimensional nonlinear spring-mass systems), here we study the corresponding issues in the simpler setting of first-order conservation law...

<p style='text-indent:20px;'>We consider nonlinear scalar conservation laws posed on a network. We define an entropy condition for networks and establish $L^1$ stability, thus uniqueness, weak solutions satisfying the condition. apply standard finite volume methods show stability convergence to unique solution, establishing existence of solution in process. Both our stability/uniqueness t...