The Numerical Treatment of a Simple Hydrodynamical Shock by the Von Neumann-Richtmyer Method

When the differential equations describing the flow of compressible fluids are derived under the assumptions that (1) forces in the fluid are due only to variations in pressure and (2) that the entropy of any volume element remains constant, it can be shown mathematically that these differential equations cannot have continuous solutions under all circumstances. If we adopt the notion of “shock discontinuities” in these solutions, the differential equations describing the flow in continuous regions together with conditions expressing the laws of conservation across the discontinuities suffice to completely determine the flow. An alternative procedure is to use a method, develop recently by J. Von Neumann and R. D. Richtmyer [3], which introduces frictional forces by the way of inclusion of a pseudo-viscosity term in differential equations. When the latter method is used, the discontinuities no longer exist. Instead, we have narrow regions over which each hydrodynamic function assumes the form of a continuous curve resembling a portion of a sine wave. The Von Neumann-Richtmyer method is much simpler to use when solving the equations by numerical procedures. In the application of numerical methods, the former method becomes almost impossible to use for complicated flows with more discontinuity, since actual applications of the conditions across these discontinuities is quite difficult and lengthy