Algebraic Flux Correction II Compressible Flow Problems

Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCT-like algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity operators, the choice of variables to be limited, and the transformation of antidiffusive fluxes. An a posteriori control mechanism is implemented to make the limiter failsafe. The numerical treatment of initial and boundary conditions is discussed in some detail. The initialization is performed using an FCT-constrained L2 projection. The characteristic boundary conditions are imposed in a weak sense, and an approximate Riemann solver is used to evaluate the fluxes on the boundary. We also present an unconditionally stable semi-implicit time-stepping scheme and an iterative solver for the fully discrete problem. The results of a numerical study indicate that the nonlinearity and non-differentiability of the flux limiter do not inhibit steady state convergence even in the case of strongly varying Mach numbers. Moreover, the convergence rates improve as the pseudo-time step is increased.