The initial data of a general Riemann problem is constant along radial directions from an origin and it is piecewise constant as a function of angle. In this thesis, the special case with initial data of piecewise constant solutions joined by four forward rarefaction waves is considered. A transonic shock is discovered through refined numerical simulations and generalized characteristic analysis, and the mathematical mechanism of shock formation for four rarefactions cases is clarified. To our best knowledge, shock formation in a rarefaction wave-only 2D Riemann problem, had not been previously observed. In this thesis, front tracking method is implemented as a numerical method based on the Riemann solution and Glimm's method; the generalized characteristic analysis method is developed into numerical characteristic analysis method in order to catch the weak transonic shock. This method is totally different from the routine one.

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