A numerical procedure for shock problem using artificial heat conduction

A method for automatically taking into account the shock discontinuities in the flow problems whenever and wherever they occur was given by Von Neumann and Richtmyer by introducing an artificial viscosity term in the momentum and energy equations. In this paper, an alternative mechanism of artificial heat conduction is proposed. This alters only the energy equation and satisfies all conditions, namely (i) the altered equations possess a continuous solution, (ii) the thickness of the shook is everywhere of the same order of interval length Δx, used in the numerical computation and is independent of the shock strength and the properties of the undisturbed medium, (iii) the dissipative mechanism is effective only in the shock layer and (iv) the Rankine-Hugoniot conditions hold when all the dimensions characterising the flow are large compared to shock thickness, as does artificial viscosity term. The stability conditions of the differential equations and the difference equations are almost the same. Numerical results for one dimensional piston problem with the piston moving with a constant speed, show excellent agreement with the exact solution