Introduction to Finite Volume Techniques in Computational Fluid Dynamics

The basic laws of fluid dynamics are conservation laws; they are statements that express the conservation of mass, momentum and energy in a volume closed by a surface. Only with the supplementary requirement of sufficient regularity of the solution can these laws be converted into partial differential equations. Sufficient regularity cannot always be guaranteed. Shocks form the most typical flow situation with a discontinuous flow field. In case discontinuities occur, the solution of the partial differential equations is to be interpreted in a weak form, i.e. as a solution of the integral form of the equations. For example, the laws governing the flow through a shock, i.e. the Hugoniot—Rankine laws, are combinations of the conservation laws in integral form. It is clear that for a correct representation of shocks, also in a numerical method, these laws have to be respected.

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