On Some Numerical Schemes for Transonic Flow Problems

Abstract

Abstract.New second-order accurate finite difference approximations for a class of nonlinear PDE's of mixed type, which includes the 2D Low Frequency Transonic Small Disturbance equation (TSD) and the 2D Full Potential equation (FP), are presented.For the TSD equation, the scheme is implemented via a time splitting algorithm; the inclusion of flux limiters keeps the total variation nonincreasing and eliminates spurious oscillations near shocks. Global Linear Stability, Total Variation Diminishing and Entropy Stability results are proven. Numerical results for the flow over a thin airfoil are presented. Current techniques used to solve the TSD equation may easily be extended to second-order accuracy by this method.For the FP equation, the new scheme requires no subsonic/supersonic switching and no numerical flux biasing. Global Linear Stability for all values of the Mach number is proven. Introduction.Recently, a number of new shock-capturing finite difference approximations for solving scalar conservation law nonlinear partial differential equations in several space dimensions have been constructed and applied to solve numerically the equations of inviscid compressible flows of aerodynamics. Those partial differential equations are, in the time-independent (steady) case, of mixed type, i.e., their type changes from elliptic to hyperbolic as the flow regime changes from subsonic to supersonic and vice versa.In this paper, we present some new shock-capturing finite difference approximations for solving scalar conservation laws. Our new schemes have the following properties:(i) second-order accuracy throughout the computational domain; (ii) global linear stability in all elliptic and all hyperbolic regions; (iii) sharp steady discrete shock solutions; (iv) total variation nonincreasing property of the approximate solutions; (v) entropy stability, at least in some cases, i.e., the approximate solutions satisfy a discrete entropy condition consistent with the differential entropy condition of