Error Bounds for Glimm Difference Approximations for Scalar Conservation Laws

We derive error bounds for the Glimm difference approximation to the solution of a genuinely nonlinear scalar conservation law with $\text {BV}$ initial data. We show that the ${L^1}$ error is bounded by $O(\Delta {x^{1/6}}|\log \Delta x|)$ in the general case, and by $O(\Delta {x^{1/2}}|\log \Delta x|)$ for a generic class of piecewise constant data.