A locally quadratic Glimm functional and sharp convergence rate of the Glimm scheme for nonlinear hyperbolic systems

TL;DR

This paper establishes a sharp convergence rate for the Glimm scheme applied to nonlinear hyperbolic systems with small initial total variation, using a novel locally quadratic Glimm functional that accounts for wave interactions.

Contribution

It introduces a new Glimm functional combining cubic and quadratic terms to improve convergence rate analysis for hyperbolic systems with specific degeneracy conditions.

Findings

Achieves sharp error estimates for Glimm solutions under certain degeneracy conditions.

Extends classical convergence results to systems with linearly degenerate manifolds.

Employs an adapted wave tracing method to analyze wave interactions.

Abstract

Consider the Cauchy problem for a strictly hyperbolic, $N\times N$ quasilinear system in one space dimension $$ u_t+A(u) u_x=0,\qquad u(0,x)=\bar u(x), \eqno (1) $$ where $u \mapsto A(u)$ is a smooth matrix-valued map, and the initial data $\overline u$ is assumed to have small total variation. We investigate the rate of convergence of approximate solutions of (1) constructed by the Glimm scheme, under the assumption that, letting $\lambda_k(u)$, $r_k(u)$ denote the $k$-th eigenvalue and a corresponding eigenvector of $A(u)$, respectively, for each $k$-th characteristic family the linearly degenerate manifold $$ \mathcal{M}_k \doteq \big\{u\in\Omega : \nabla\lambda_k(u)\cdot r_k(u)=0\big\} $$ is either the whole space, or it is empty, or it consists of a finite number of smooth, $N-1$-dimensional, connected, manifolds that are transversal to the characteristic vector field $r_k$. We introduce a Glimm type functional which is the sum of the cubic interaction potential defined in \cite{sie}, and of a quadratic term that takes into account interactions of waves of the same family with strength smaller than some fixed threshold parameter. Relying on an adapted wave tracing method, and on the decrease amount of such a functional, we obtain the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems satisfying the classical Lax assumptions of genuine nonlinearity or linear degeneracy of the characteristic families.