Date of Award

Summer 1991

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematics & Statistics


Computational and Applied Mathematics

Committee Director

Charlie H. Cooke

Committee Member

John H. Heinbockel


The numerical investigation of shock phenomena in gas or liquid media where a specifying relation for internal energy is absent poses special problems. Classically, for gas dynamics the usual procedure is to employ a splitting scheme to remove the source terms from the Euler equations, then up-wind biased shock capturing algorithms are built around the Riemann problem for the system which remains. However, in the case where the Euler equations are formulated in the term of total enthalpy, a technical difficulty associated with equation splitting forces a pressure time derivative to be treated as a source term. This makes it necessary to pose the central Riemann problem for a system where one equation is not a conservation law. In the present research, it is established that successful upwind-biased shock capturing schemes can be applied to the pseudo-conservative system. A shock capturing method developed for this purpose is applied to solving the Riemann problem for pure water with general (UNESCO (8)) equation of state.

A second objective of this research is the development of front tracking methods which possess sub-grid resolution capability. One envisions here continuous tracking of the front, as opposed to discrete (one grid point to the next) tracking, such as is provided by the random choice method. A front tracking scheme employing near-front cells which continuously evolve with time is developed. This scheme is applied to the problem of tracking a material interface in the underwater explosion problem. The Riemann problem for water and for a gas-water interface is analyzed, as a vehicle for applying the Godunov method to shock phenomena in liquid media. A by-product of this study is a mapping theorem which establishes a certain correspondence between solutions of the Riemann problem(s) for the ideal gas and the ideal water (modified-Tait equation of state). As a result, present codes which solve the Riemann problem for gases can readily be adapted for use with water.


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