Multidimensional FEM‐FCT schemes for arbitrary time stepping

Abstract

SUMMARYThe ux-corrected-transport paradigm is generalized to ÿnite-element schemes based on arbitrary time stepping. A conservative ux decomposition procedure is proposed for both convective and di usive terms. Mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory loworder method is constructed by elimination of negative o -diagonal entries of the discrete transport operator. The linearization of source terms and extension to hyperbolic systems are discussed. Zalesak's multidimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof of positivity is provided. The treatment of non-linearities and iterative solution of linear systems are addressed. The performance of the new algorithm is illustrated by numerical examples for the shock tube problem in one dimension and scalar transport equations in two dimensions.

Efficient solution techniques for implicit finite element schemes with flux limiters

“…These restrictions have led to the development of a generalized FEM-FCT methodology introduced by Kuzmin and Turek [9] and refined by Kuzmin et al [10][11][12]. Flux correction of FCT type is readily applicable to Galerkin schemes with a consistent mass matrix.…”

“…A detailed description of this so-called algebraic flux correction (AFC) paradigm can be found in [9][10][11][12][13][14][15]17]. As a model problem, consider a stationary conservation law for a scalar quantity u whereby f is a generic and possibly nonlinear flux function…”

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

“…The new error indicators were applied to algebraic flux correction [14][15][16][17][18][19] schemes which were successfully equipped with grid adaptivity. The highly unstructured meshes resulting from local mesh refinement/coarsening call for fully implicit discretizations which are unconditionally stable.…”

“…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.…”