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This IMA Volume in Mathematics and its Applications Oscillation Theory, Computation, and Methods of Compensated Compactness represents the proceedings of a workshop which was an integral part of the 1984-85 IMA program on CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EQUATIONS. We are grateful to the Scientific Committee: J. L. Ericksen D. Kinderlehrer H. Brezis C. Dafermos for their dedication and hard work in developing an imaginative, stimulating, and productive year-long program. George R. Sell Hans Weinberger PREFACE Historically, one of the most important prohlems in continuum mechanics has been the treatment of nonlinear hyperbolic systems of conservation laws. Thp. importance of these systems lies in the fact that the underlyinq equ~tions of mass, momentum, and energy are descrihed by conservation laws. Their nonlinearity and hyperbolicity are consequences of some cornmon constitutive relations, for example, in an ideal gas. The I. M. A. Workshop on "Osci 11 at i on theory. computat i on, and methods of com pensated compactness" brought together scientists from both the analytical and numerical sides of conservation law research. The goal was to examine recent trends in the investigation of systems of conservation laws and in particular to focus on the roles of dispersive and diffusive limits for singularily perturbed conservation laws. Special attention was devoted to the new ideas of compen sated compactness and oscillation theory.
List of contents
Convection of Microstructures by Incompressible and Slightly Compressible Flows.- Oscillations in Solutions to Nonlinear Differential Equations.- Geometry and Modulation Theory for the Periodic Nonlinear Schrodinger Equation.- On High-Order Accurate Interpolation for Non-Oscillatory Shock Capturing Schemes.- On the Weak Convergence of Dispersive Difference Schemes.- Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws.- On the Construction of a Modulating Multiphase Wavetrain for a Perturbed KdV Equation.- Evidence of Nonuniqueness and Oscillatory Solutions in Computational Fluid Mechanics.- Very High Order Accurate TVD Schemes.- Convergence of Approximate Solutions to Some Systems of Conservative Laws: A Conjecture on the Product of the Riemann Invariants.- Applications of the Theory of Compensated Compactness.- A General Study of a Commutation Relation given by L. Tartar.- Interrelationships among Mechanics, Numerical Analysis, Compensated Compactness, and Oscillation Theory.- The Solution of Completely Integrable Systems in the Continuum Limit of the Spectral Data.- Stability of Finite-Difference Approximations for Hyperbolic Initial-Boundary-Value Problems.- Construction of a Class of Symmetric TVD Schemes.- Information About Other Volumes in this Program.
