Error Bounds for a Deterministic Version of the Glimm Scheme

Abstract.

Consider the hyperbolic system of conservation laws $u_t+F(u)_x=0$. Let u be the unique viscosity solution with initial condition $u(0,x)=\bar u(x)$, and let u ^ε be an approximate solution constructed by the Glimm scheme, corresponding to the mesh sizes $\Delta x$, $\Delta t =O(\Delta x)$. With a suitable choice of the sampling sequence, we prove the estimate $\big\| u^\ve(t,\cdot)-u(t,\cdot)\big\|_{{\bf L}^1} %{\strut\L^1}=o(1)\cdot \sqrt{\Delta x} \big|\ln (\Delta x)\big|.$