There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one–parameter Hamiltonian symmetries admitted by the PDEs [1]. The same result holds for semidiscrete Hamiltonian equations [2]. In this paper we consider semidiscrete canonical Hamiltonian equations. Using symmetries, we find conservation laws for...

A complete conservation law classification is given for nonlinear telegraph (NLT) systems with respect to multipliers that are functions of independent and dependent variables. It turns out that a very large class of NLT systems admits four nontrivial local conservation laws. The results of this work are summarized in tables which display all multipliers, fluxes and densities for the correspond...

A simple conservation law formula for field equations with a scaling symmetry is presented. The formula uses adjoint-symmetries of the given field equation and directly generates all local conservation laws for any conserved quantities having non-zero scaling weight. Applications to several soliton equations, fluid flow and nonlinear wave equations, Yang-Mills equations and the Einstein gravita...

A survey of several nite diierence methods for systems of nonlinear hyperbolic conservation laws, J.

The Riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed. The inhomogeneity is modeled by a linearly degenerate field. The solution ofthe Riemann problem determines the nature of wave interactions, and thus the Riemann problem serves as a canonical form for nonlinea...

This is a summary of ve lectures delivered at the CIME course on "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations" held in Cetraro, Italy, on June 1997. Following the introductory lecture I | which provides a general overview of approximate solution to nonlinear conservation laws, the remaining lectures deal with the speciics of four complementing topics: Lecture II. Finite-d...

A new class of high-order numerical methods for approximating nonlinear conservation laws is described (entropy viscosity method). The novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand. This new approach does not use any flux or slope limiters, applies to equations or systems supplemented with one or more entr...

A central problem in computational fluid dynamics is the development of the numerical approximations for nonlinear hyperbolic conservation laws and related time-dependent problems governed by additional dissipative and dispersive forcing terms. Entropy stability serves as an essential guideline in the design of new computationally reliable numerical schemes. My dissertation research involves a ...

The concept of multiresolution-based adaptive DG schemes for nonlinear one-dimensional hyperbolic conservation laws has been developed and investigated analytically and numerically in N. Hovhannisyan, S. Müller, R. Schäfer, Adaptive multiresolution Discontinuous Galerkin Schemes for Conservation Laws, Math. Comp., 2013. The key idea is to perform a multiresolution analysis using multiwavelets o...

Using standard calculus, explicit formulas for one-, twoand three-dimensional homotopy operators are presented. A derivation of the one-dimensional homotopy operator is given. A similar methodology can be used to derive the multi-dimensional versions. The calculus-based formulas for the homotopy operators are easy to implement in computer algebra systems such as Mathematica, Maple, and REDUCE. ...