Vanishing Viscosity Approximation to Hyperbolic Conservation Laws

We study high order convergence of vanishing viscosity approximation to scalar hyperbolic conservation laws in one space dimension. We prove that, under suitable assumptions, in the region where the solution is smooth, the viscous solution admits an expansion in powers of the viscosity parameter ε. This allows an extrapolation procedure that yields high order approximation to the non-viscous limit as ε → 0. Furthermore, an integral across a shock also admits a power expansion of ε, which allows us to construct high order approximation to the location of the shock. Numerical experiments are presented to justify our theoretical findings.