A Discrete Algorithm for General Weakly Hyperbolic Systems

TL;DR

This paper establishes well-posedness and stability results for weakly hyperbolic systems with variable coefficients, and analyzes the spectral Crank-Nicholson scheme's accuracy and computational cost for approximate solutions.

Contribution

It provides the first analysis of the spectral Crank-Nicholson scheme for weakly hyperbolic systems, including stability, error estimates, and polynomial cost bounds.

Findings

Unique Gevrey regular solutions under certain conditions

Stability and error estimates for the spectral Crank-Nicholson scheme

Polynomial growth in computational cost for desired accuracy

Abstract

This paper studies the Cauchy problem for variable coefficient weakly hyperbolic first order systems of partial differential operators. The hyperbolicity assumption is that for each $t, x$ the principal symbol is hyperbolic. No hypothesis is imposed on lower order terms. For coefficients and Cauchy data sufficiently Gevrey regular the Cauchy problem has a unique sufficiently Gevrey regular solution. We prove stability and error estimates for the spectral Crank-Nicholson scheme. Approximate solutions can be computed with accuracy $epsilon$ in the supremum norm with cost growing at most polynomially in $epsilon^{-1}$. The proofs use the symmetrizers from [2].