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Tricomi's Equation and Condition for Shock-Free Flow

PDE and Applied Math Seminar

Speaker: Jean-Jacques Chattot, Mechanical & Aerospace Engineering, UC Davis
Location: 1147 MSB
Start time: Tue, Nov 29 2011, 3:10PM

A Dirichlet boundary value problem is investigated for the Tricomi equation, in relation to shock-free solutions, their uniqueness and their numerical computation. The study of uniqueness and existence or rather non-existence of shock-free solutions by Morawetz was based on the formulation of a Goursat type problem, in which only part of the streamline data is given in the hodograph plane, the missing part corresponding to the arc intercepted by the two characteristic lines coming from a point on the sonic line. In the physical plane, on the other hand, the complete profile shape determines the solution. To reconcile the formulations in both the physical and the hodograph planes, a Dirichlet boundary value problem is proposed in the hodograph plane. The requirement for such a problem to be well-posed is that the streamline that represents the profile be “sub-characteristic”, a condition that is shown to yield a shock-free flow. A class of symmetrical, solutions to Tricomi's equation is presented and we discuss conditions for shock-free flow and for the need of a shock to regularize the solution. As a first step, such a formulation is applied to the wave equation and it is shown that, provided the boundary is “sub-characteristic”, a unique solution exists and is obtained numerically in a simple example. Along the same line, the uniqueness of the Dirichlet problem as well as the computation, to second order accuracy, of the analytic solution of the Tricomi equation presented earlier, are demonstrated.