Steady Shock Tracking, Newton’s Method, and the Supersonic Blunt Body Problem

The steady shock tracking method, which combines shock tracking and Newton’s method, is applied to the axisymmetric supersonic blunt body problem. The formulation is in conservation form and uses the constant total enthalpy condition to reduce the number of unknowns at each finite difference mesh point. On a transformed computational grid where the bow shock is a coordinate line, the discrete physical shock locations appear explicitly as unknowns in a set of finite difference equations which couples them to the other unknowns. The space-time characteristic compatibility conditions for the associated time-dependent problem are used in formulating the boundary conditions for the steady problem. The resulting system is solved using various modifications of Newton’s method. Experiments are repeated with three linear system solvers whose efficiency is compared. The computed results for flow over a sphere are economically obtained and agree well with experiment. Continuation of the solutions with respect to some physical parameters is explored, and multiple solutions of one variation of our finite difference system are displayed.