Computation of steady shocks by second-order finite-difference schemes

Abstract:

The computational stability of steady shocks which satisfy the entropy condition is considered for the scalar conservation law \[ \frac {{\partial u}}{{\partial t}} + \frac {\partial }{{\partial x}}\left ( {\frac {1}{2}{u^2}} \right ) = 0.\] It is shown that the computation of the pure initial value problem by Lax-Wendroff type schemes approaches a steady state if the initial data satisfies a specified condition, and that this condition is always satisfied for the corresponding initial-boundary value problem with a finite number of grid points. The effect of machine accuracy on the influence of the boundaries on the error near the shock is also discussed.