High resolution finite difference methods have been successfully employed in solving Hamilton Jacobi equations, and numerical Hamiltonians are as important as numerical fluxes for solving hyperbolic conservation laws using the finite volume methodology. Though different numerical Hamiltonians have been suggested in the literature, only some simple ones such as the Lax-Friedrichs Hamiltonian are...

This paper studies the boundary layers that generally arise in approximations of the entropy discontinuous solutions to the initial-boundary value problem associated with a nonlinear hyperbolic system of conservation laws. We consider the vanishing viscosity method and several nite diierence schemes (Lax-Friedrichs type schemes, Godunov scheme). Assuming solely uniform L 1 bounds and for entrop...

Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydroand magnetohydrodynamic simulations, AMR is used in combination with several shock-capturing, conservative discretization schemes. Solution accuracy and execution times are compared with st...

This paper studies the boundary layers that generally arise in approximations of the entropy discontinuous solutions to the initial-boundary value problem associated with a nonlinear hyperbolic system of conservation laws. We consider the vanishing viscosity method and several nite di erence schemes (Lax-Friedrichs type schemes, Godunov scheme). Assuming solely uniform L1 bounds and for entropy...

6 Friedrichs, K. O., Mathematical Aspects of the Quantum Theory of Fields (New York: Interscience, 1953). 7Friedrichs, K. O., Lectures, Summer Conference on Mathematics and Modern Physical Theories, Boulder (1960). 8 Schwartz, J., Mass Renormalization and Spectral Shifts, N.Y.U. Research Report (1960). 9 Friedrichs, K. O., "Spectral theory of operators in Hilbert space," N.Y.U. lecture notes (1...

The study of pseudodifferential operators emerged in the 1960’s, having its origins in the study of singular integro-differential operators. In fact, Friedrichs and Lax coined the term “pseudodifferential operator” in their 1965 paper entitled “Boundary Value Problems for First Order Operators”. Since that time, pseudodifferential operators have proved useful in many arenas of modern analysis a...

In this paper, we focus on the application and exemplification of the approach developed in the companion paper. This approach is found to be useful in the construction of stable and monotone central difference schemes for hyperbolic systems. The stability of several versions of the central scheme developed in the companion paper is proven. Sufficient conditions for the stability as well as nec...

This paper considers a fluid dynamic traffic flow model appended with a closure linear velocity-density relationship which provides a first order hyperbolic partial differential equation (PDE) and is treated as an initial boundary value problem (IBVP). We consider the boundary value in such a way that one side of highway treat like there is a traffic controller at that point. We present the ana...

We present three-dimensional central finite volume methods for solving systems of hyperbolic equations. Based on the Lax–Friedrichs and Nessyahu–Tadmor one-dimensional central finite difference schemes, the numerical methods we propose involve an original and a staggered grid in order to avoid the resolution of the Riemann problems at the cell interfaces. The cells of the original grid are Cart...

The approximation of shock waves by nite diierence schemes is considered. This question has been investigated by many authors, but mainly under some restrictions on discrete wave speeds. Basic works are due to Majda and Ralston (rational speed) and, more recently, to Liu and Yu (Diophantine speed). The main purpose of the present work is to obtain shock prooles of arbitrary speed for rather gen...