Convergence rates of monotone schemes for conservation laws for data with unbounded total variation

TL;DR

This paper establishes convergence rates for monotone schemes solving conservation laws with initial data of unbounded total variation, especially when the data is H"older continuous with an exponent greater than 1/2, supported by numerical verification.

Contribution

It provides new convergence rate results for monotone schemes applied to conservation laws with unbounded variation initial data, extending previous understanding.

Findings

Convergence rates are proven for H"older continuous initial data with exponent > 1/2.

For strictly Lip^+ stable schemes, convergence is shown for all positive H"older exponents.

Numerical experiments confirm the theoretical convergence rates.

Abstract

We prove convergence rates of monotone schemes for conservation laws for H\"older continuous initial data with unbounded total variation, provided that the H\"older exponent of the initial data is greater than $1/2$. For strictly $\mathrm{Lip}^+$ stable monotone schemes, we prove convergence for any positive H\"older exponent. Numerical experiments are presented which verify the theory.