By Joseph L. Steger (auth.), M. Y. Hussaini, A. Kumar, M. D. Salas (eds.)
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Extra info for Algorithmic Trends in Computational Fluid Dynamics
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Valli: An existence theorem for compressible viscous flow. Mat. It. Anal. Finz. Appl. 18-C, 317-325, 1981. [34J C. Bernardi, O. Pironneau: Existence And Approximation Of Compressible Viscous Isothermal Stationary Flows. Note CRAS, 1990.  O. Pironneau, J. Rappaz: Numerical Analysis for Compressible Isentropic Viscous Flows. IMPACT. 1989. [36J L. In Systems of Nonlinear PDEs, J. Ball ed. NATO series AS!. C. Reidel Publishing Co. 1983. [37J J. A. V. C. Ciccoli: Developments in hypersonic reactive flow computations.
The modern versions, based on Riemann solvers, keep the equations in conservation form and therefore, at least in one-dimensional problems, the shockspeed is automatically correct. In more space dimensions the theory is not satisfactory, because the methods in use are often based on dimensional splitting, which can only be justified for smooth solutions. There has been an explosion of powerful methods. Without them many problems, especially with strong shocks, are difficult to solve. Still, the theory for systems in two or three space dimensions is not yet completely developed.