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

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.