Calculation of shocked one-dimensional flows on abruptly changing grids by mathematical programming

Abstract

The steady-state inviscid and nearly inviscid Burgers' equations and the Euler equations for steady-state shocked flow in a quasi-1-dimensional nozzle are discretized by cell-centered finite differences on the cells of an arbitrarily spaced grid. inviscid and viscous terms are approximated to second order by 2-point schemes and 4-point schemes, respectively. There results an overdetermined system of nonlinear algebraic equations. This system is solved by a mathematical-programming procedure that minimizes a weighted sum of the absolute values of the residuals (the l 1 norm of the vector of residuals). In this algorithm, which is nonconservative, no upwinding, switches, arbitrary constants, or heuristic quantities are used. The artificial viscosity used to solve the inviscid Burgers' equation is small (−10 −15 u″). No artificial viscosity of any kind is used for the Euler equations. The numerical solutions of both the viscous and the inviscid problems are accurate and nonoscillatory on grids with abrupt refinements in mesh length by factors as high as 10 4. Shocks are invariably captured in one cell and this cell is rarely more than three cells away from the cell in which the physical shock occurs.