Godunov-Type Shcemes For Pressure Gradient Equations And Isentropic Fluid Flows

This dissertation firstly uses the classical Godunov scheme to compute the numerical solution of one dimensional Eulerian pressure gradient equations. To our surprise, the Godunov scheme can not be performed well for this system when the Riemann solution contains a weak backward rarefaction wave and a strong forward shock. To investigate the "numerical phenomenon", we analyze the Riemann solver of this system and the convergence of Newton-Raphson iteration. Furthermore, we compute the Riemann solution by using a bisection method combined with the phase-plane analysis.The generalized Riemann problem(GRP) scheme for the pressure gradient equations is also derived and used to compute this system. Our main purpose is to see if this second order(both in space and time) Godunov-type scheme can avoid above "numerical phenomenon". We get our conclusion that GRP scheme for the pressure gradient equations can not be performed well either when the Riemann solutions contain weak simple waves, according to the numerical experiments.To study above "numerical phenomenon" further, both the Godunov scheme and GRP scheme for one dimensional Lagrangian isentropic fluid flows, whose structure of eigenvalues is similar to that of the pressure gradient equations, are derived and used to compute the numerical solution. The numerical experiments imply that the Godunov-type schemes are performed well for strong waves and some weak waves, but they can not be performed well for some other cases in which the Riemann solutions contain weak waves.