Implicit Flux-Corrected Transport Algorithm for Finite Element Simulation of the Compressible Euler Equations

Abstract

Even today, the accurate treatment of convection-dominated transport problems remains a challenging task in numerical simulation of both compressible and incompressible flows. The discrepancy arises between high accuracy and good resolution of singularities on the one hand and preventing the growth and birth of nonphysical oscillations on the other hand. In 1959 it was proven [6], that linear methods are restricted to be at most first order if they are to preserve monotonicity. Thus, the use of nonlinear methods is indispensable to overcome smearing by numerical diffusion without sacrificing important properties of the exact solution such as positivity and monotonicity. The advent of the promising methodology of flux-corrected transport (FCT can be traced back to the pioneering work of Boris and Book [3]. Even though their original FCT algorithm named SHASTA was a rather specialized one-dimensional finite difference scheme, the cornerstone for a variety of high-resolution schemes was laid. Strictly speaking, the authors recommended using a high-order discretization in regions of smooth solutions and switching to a low-order method in the vicinity of steep gradients. This idea of adaptive toggling between methods of high and low order was dramatically improved by Zalesak [3] who proposed a multi-dimensional generalization applicable to arbitrary combinations of high- and low-order discretizations but still remaining in the realm of finite differences. This barrier was first crossed by Parrott and Christie [21] who settled the idea of flux-correction in the framework of finite elements. Finally, FEM-FCT reached maturity by the considerable contributions of Löhner and his coworkers [16], [17]. Beside the classical formulation of Zalesak’s limiter in terms of element contributions, an alternative approach is available limiting the fluxes edge-by-edge [26], [27].