An existence theorem in the large is obtained for the Riemann problem for nonlinear systems of conservation laws. Our principal assumptions are strict hyperbolicity, genuine nonlinearity in the strong sense, and the existence of a convex entropy function. The entropy inequality is used to obtain an a priori estimate of the strengths of the shocks and refraction waves forming a solution; existen...

We employ a nonlinear anisotropic di usion operator like the ones used as a means of ltering and edge enhancement in image processing, in numerical methods for conservation laws. It turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher-order dissipative terms. In particular, we stabilize the well-known Lax-Wendro formula by means of...

A mathematical formulation of conservation and of balance laws with random input data, specifically with random initial conditions, random source terms and random flux functions, is reviewed. The concept of random entropy solution is specified. For scalar conservation laws in multi-dimensions, recent results on the existence and on the uniqueness of random entropy solutions with finite variance...

This chapter highlights recent developments concerning adaptive wavelet methods for time dependent and stationary problems. The first problem class focusses on hyperbolic conservation laws where wavelet concepts exploit sparse representations of the conserved variables. Regarding the second problem class, we begin with matrix compression in the context of boundary integral equations where the k...

In this paper, a Chebyshev–Legendre spectral viscosity (CLSV) method is developed for nonlinear conservation laws with initial and boundary conditions. The boundary conditions are dealt with by a penalty method. The viscosity is put only on the high modes, so accuracy may be recovered by postprocessing the CLSV approximation. It is proved that the bounded solution of the CLSV method converges t...

We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate ...

Kinematical conservation laws (KCL) is a system of conservation laws governing the evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [1] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. T...

Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in Mathematica. T...

In this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in L2, provided that the sums of the initial...

We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain’s bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate ...