The equations for the Riemann problem for a perfect gas or constant covolume equation of state are rewritten in terms of three similarity parameters and a weak dependence on the ratio of specific heats of the gases. These parameters are used to demonstrate the errors in the solutions of existing linearized Riemann solvers and to develop a new solver that has the simplicity of the linearized solvers but that is more accurate. Among nonlinear solvers, the two-expansion approximation is found to be very accurate, except for flows involving strong shock waves. An equivalent two-shock approximation is developed that is very accurate for converging flows. It is combined with the two-expansion approximation to give an approximation that has bounded error for all initial conditions. More accurate or exact solutions to the Riemann problem require iterative methods. Faster exact solvers are obtained by using the new approximations as starting values, by reformulating the iteration technique for faster convergence, and by using a "look-up" table to speed up part of the calculation. Average solution times are reduced to about two-thirds of the previous fastest method.
The Riemann problem concerns the development of the flow at the boundary between two constant flow states. This initial flow is described by the pressure, density, velocity and the ratio of specific heats either side of a discontinuity; a total of 8 parameters. The reduction of the Riemann problem effectively to 3 parameters, allows a assessment of the accuracy of an approximation for all conditions that can occur. The analysis is used to develop a 2-Shock 2-Expansion approximation with bounded error limits. This ensures that when the approximation is used in a calculation method, the method will not lose accuracy or crash when unlikely conditions which cause large errors in the Riemann solution are encountered. The approximation is also used with other techniques to speed up the interation of the exact solution.
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Last amended: Nov 2012.