A Shock Capturing Method

Strong discontinuities, or shocks in continua are a result of external dynamic loads. On the shock surface the conservation laws take the form of nonlinear algebraic equations for jumps across the shock. Entropy jumps across a strong discontinuity, and just this jump differs shocks from waves where the quantities vary continuously. In the heterogeneous difference schemes, the shock is treated as a layer of a finite thickness comparable with the cell size. This property of finite-difference schemes was called distraction. Since the state behind a shock is related to the state before it by the Hugoniot, in the distraction region there must act a mechanism that increases entropy. The physical viscosity and heat conductivity in continuum mechanics equations do not make it unnecessary to introduce a shock surface and hence cannot make the distraction length comparable with a few cells of the difference mesh. The paper considers a number of finite difference schemes where energy dissipation in the distraction region is defined by equations which are valid on the shock surface.