Convergence properties of detonation simulations

We present a high-resolution convergence study of detonation initiated by a temperature gradient in a stoichiometric hydrogen-oxygen mixture using the Pencil Code and compare with a code that employs a fifth order weighted essentially non-oscillating (WENO) scheme. With Mach numbers reaching 10-30, a certain amount of shock viscosity is needed in the Pencil Code to remove or reduce numerical pressure oscillations on the grid scale at the position of the shock. Detonation is found to occur for intermediate values of the shock viscosity parameter. At fixed values of this parameter, the numerical error associated with those small wiggles in the pressure profile is found to decrease with decreasing mesh width $\delta x$ like $\delta x^{-1.4}$ down to $\delta x=0.2\mu$m. With the WENO scheme, solutions are smooth at $\delta x=10\mu$m, but no detonation is obtained for $\delta x=5\mu$m. This is argued to be an artifact of a decoupling between pressure and reaction fronts.