Errors when shock waves interact due to numerical shock width

A simple test problem proposed by Noh, a strong shock reflecting from a rigid wall, demonstrates a generic problem with numerical shock capturing algorithms at boundaries that Noh called 'excess wall heating'. We show that the same type of numerical error occurs in general when shock waves interact. The underlying cause is the non-uniform convergence to the hyperbolic solution of the inviscid limit of the solution to the PDE's with viscosity. The error can be understood from an analysis of the asymptotic solution. For a propagating shock, there is a difference in the total energy of the parabolic wave relative to the hyperbolic shock. Moreover, the relative energy depends on the strength of the shock. The error when shock waves interact is due to the difference in the relative energies between the incoming and outgoing shock waves. It is analogous to a phase shift in a scattering matrix. A conservative differencing scheme correctly describes the Hugoniot jump conditions for a steady propagating shock. Therefore, the error from the asymptotics occurs in the transient when the waves interact. The entropy error that occurs in the interaction region remains localized but does not dissipate. A scaling argument shows that as the viscosity coefficient goes to zero, the error shrinks in spatial extention but is constant in magnitude. Noh's problem of the reflection of a shock from a rigid wall is equivalent to the symmetric impact of two shock waves of the opposite family. The asymptotic argument shows that the same type of numerical error would occur when the shocks are of unequal strength. Thus, Noh's problem is indicative of a numerical error that occurs when shocks interact due to the numerical shock width.