Implicit Finite Element Discretizations based on the Flux-Corrected Transport Algorithm

The flux-corrected transport (FCT) methodology is generalized to implicit finite element schemes and applied to the Euler equations of gas dynamics. For scalar equations, a local extremum diminishing scheme is constructed by adding artificial diffusion so as to eliminate negative off-diagonal entries from the high-order transport operator. To obtain a nonoscillatory low-order method in the case of hyperbolic systems, the artificial viscosity tensor is designed so that all off-diagonal blocks of the discrete Jacobians are rendered positive semi-definite. Compensating antidiffusion is applied within a fixed-point defect correction loop so as to recover the high accuracy of the Galerkin discretization in regions of smooth solutions. All conservative matrix manipulations are performed edge-by-edge which leads to an efficient algorithm for the matrix assembly.