Algebraic Flux Correction I. Scalar Conservation Laws

Abstract

This chapter is concerned with the design of high-resolution finite element schemes satisfying the discrete maximum principle. The presented algebraic flux correction paradigm is a generalization of the flux-corrected transport (FCT) methodology. Given the standard Galerkin discretization of a scalar transport equation, we decompose the antidiffusive part of the discrete operator into numerical fluxes and limit these fluxes in a conservative way. The purpose of this manipulation is to make the antidiffusive term local extremum diminishing. The available limiting techniques include a family of implicit FCT schemes and a new linearity-preserving limiter which provides a unified treatment of stationary and time-dependent problems. The use of Anderson acceleration makes it possible to design a simple and efficient quasi-Newton solver for the constrained Galerkin scheme. We also present a linearized FCT method for computations with small time steps. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations.

On the Use of Slope Limiters for the Design of Recovery Based Error Indicators

“…For a numerical scheme to be non-oscillatory, it should possess certain properties 16 , e.g., be monotone, positivity preserving or total variation diminishing (TVD).…”

“…Mass lumping can also be applied directly to equation (16) which yields an explicit formula for computing the values of the projected gradient at each nodê…”

“…In general, provisional slopes at edge midpoints can be interpolated from the nodal values obtained either from (16) or (20) making use of representation (14). For linear finite elements this corresponds to taking the mean values for each edge ij,…”

“…The bisection process continues for all adjacent triangles sharing a hanging node with the refined element until all irregular grid points have been removed from the mesh. However, longest-edge bisection is mainly designed to uphold some geometric properties of the initial mesh and, thus, may not be the best comrade for our algebraic flux correction techniques [16][17][18][19] . For each element that needs to be refined due to accuracy reasons, the propagation path solely depends on the mesh geometry and does not account for the local solution behavior.…”

“…In a series of recent publications [14][15][16][17][18][19] , an algebraic framework for the construction of high-resolution schemes for convection dominated partial differential equations was developed. The algebraic flux correction (AFC) paradigm renders a high-order discretization local extremum diminishing (LED) by a conservative elimination of negative off-diagonal entries from the discrete transport operator so as to end up with a non-oscillatory approximation of low order.…”