Eli Isaacson

Convergence of the 2×2 Godunov method for a general resonant nonlinear balance law

We introduce a generalized solution of the Riemann problem for a general resonant nonlinear balance law, and we prove the convergence of the 2 x 2 Godunov numerical method based on these solutions. In particular, we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the Riemann problem, and an interesting multiplicity of time-asymptotic wave patterns is observed and characterized.

Nonlinear resonance in systems of conservation laws

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 of the Riemann problem determines the nature of wave interactions, and thus the Riemann problem serves as a canonical form for nonlinear systems of conservation laws. Generic conditions on the fluxes are stated and it is proved that under these conditions, the solution of the Riemann problem exists, is unique, and has a fixed structure; this demonstrates that, in the above sense, resonant inhomogeneous systems generically have the same canonical form. The wave curves for these systems are only Lipschitz continuous in a neighborhood of the states where the wave speeds coincide, and so, in contrast to strictly hyperbolic systems, the implicit function theorem cannot be applied directly to obtain existence and uniqueness. Here we show that existence ...

Transitional waves for conservation laws

A new class of fundamental waves arises in conservation laws that are not strictly hyperbolic. These waves serve as transitions between wave groups associated with particular characteristic families. Transitional shock waves are discontinuous solutions that possess viscous profiles but do not conform to the Lax characteristic criterion; they are sensitive to the precise form of the physical viscosity. Transitional rarefaction waves are rarefaction fans across which the characteristic family changes from faster to slower.This paper identifies an extensive family of transitional shock waves for conservation laws with quadratic fluxes and arbitrary viscosity matrices; this family comprises all transitional shock waves for a certain class of such quadratic models. The paper also establishes, for general systems of two conservation laws, the generic nature of rarefaction curves near an elliptic region, thereby identifying transitional rarefaction waves. The use of transitional waves in solving Riemann problems...

The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems 1

The purpose of this paper is to classify the solutions of Riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws. Hyperbolic singularities play the role in the theory of Riemann problems that rest points play in the theory of ordinary differential equations: Indeed, generically, only a finite number of structures can appear in a neighborhood of such a singularity. In this, the first of three papers, the program of classification is discussed in general and the simplest structure that occurs is characterized.

Analysis of a Singular Hyperbolic System of Conservation Laws

Abstract We solve the Riemann and Cauchy problems globally for a singular system of n hyperbolic conservation laws. The system, which arises in the study of oil reservoir simulation, has only two wave speeds, and these coincide on a surface of codimension one in state space. The analysis uses the random choice method of Glimm ( Comm. Pure Appl. Math. 18 (1965), 697–715).

An explicit solution of the inverse periodic problem for Hill's equation

Let the periodic spectrum of the Hill’s operator

The structure of asymptotic states in a singular system of conservation laws

{{ - d^2 } / {dx^2 + p(x)}}

A Fully Implicit Scheme for the Barotropic Primitive Equations

have n nonzero gaps. We give explicit formulas for the isospectral manifold of operators

Numerical analysis of spectral properties of coupled oscillator Schrödinger operators. I. Single and double well anharmonic oscillators

{{ - d^2 } / {dx^2 + q(x)}}

The Structure of the Riemann Solution for Non-Strictly Hyperbolic Conservation Laws

having the same spectrum. This allows us to realize the isospectral manifold explicitly as a torus. What makes this possible is an explicit solution of the flow \[ \left. {\frac{d}{{dt}}q = \frac{d}{{dx}}\frac{\partial }{{\partial q(x)}}\Delta (\lambda ,q)} \right|_{\lambda = \mu _n (q)} \] introduced by McKean and Trubowitz, where