Obstacle problems for scalar conservation laws

Viscous Conservation Laws, Part I: Scalar Laws

Viscous conservation laws are the basic models for the dissipative phenomena. We aim at a systematic presentation of the basic ideas for the quantitative study of the nonlinear waves for viscous conservation laws. The present paper concentrates on the scalar laws; an upcoming Part II will deal with the systems. The basic ideas for scalar viscous conservation laws originated from two sources: th...

Regularity through Approximation for Scalar Conservation Laws∗

In this paper it is shown that recent approximation results for scalar conservation laws in one space dimension imply that solutions of these equations with smooth, convex fluxes have more regularity than previously believed. Regularity is measured in spaces determined by quasinorms related to the solution’s approximation properties in L1(R) by discontinuous, piecewise linear functions. Using a...

Monotone Difference Approximations for Scalar Conservation Laws

A complete self-contained treatment of the stability and convergence properties of conservation-form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that general monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous conve...

Weakly nonoscillatory schemes for scalar conservation laws

A new class of Godunov-type numerical methods (called here weakly nonoscillatory or WNO) for solving nonlinear scalar conservation laws in one space dimension is introduced. This new class generalizes the classical nonoscillatory schemes. In particular, it contains modified versions of MinMod and UNO. Under certain conditions, convergence and error estimates for WNO methods are proved.

L2 formulation of multidimensional scalar conservation laws