A nonlinear functional for general scalar hyperbolic conservation laws

A Nonlinear Flux Split Method for Hyperbolic Conservation Laws

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the [Formula: see text]-th order [Formula: see text] divided difference of the DG error in the [Fo...

On a Quadratic Functional for Scalar Conservation Laws

We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. The proof is based on the introduction of a quadratic functional Q(t), decreasing at every interaction, and such that its to...

A Note on the New Glimm Functional for General Systems of Hyperbolic Conservation Laws

For systems of hyperbolic conservation laws, a new Glimm functional was recently constructed when the linearly degenerate manifold in each characteristic field is either the whole space or it consists of a finitely many smooth and transversal manifolds of co-dimension one. This new functional leads to the neat consistency and convergence rate estimation of the Glimm scheme. In this paper, by th...

Error Estimates of Approximate Solutions for Nonlinear Scalar Conservation Laws

There has been an enormous amount of work on error estimates for approximate solutions to scalar conservation laws. The methods of analysis include matching the traveling wave solutions, [8, 24]; matching the Green function of the linearized problem [21]; weak W convergence theory [32]; the Kruzkov-functional method [19]; and the energy-like method [34]. The results on error estimates include: ...