Explicit and implicit FEM-FCT algorithms with flux linearization

A new approach to the design of flux-corrected transport (FCT) algorithms for linear/bilinear finite element approximations of convection-dominated transport problems is pursued. The raw antidiffusive fluxes are linearized about an intermediate solution computed by a positivity-preserving low-order scheme. By virtue of this linearization, the costly evaluation of correction factors needs to be performed just once per time step, and no nonlinear algebraic systems need to be solved if the governing equation is linear. Furthermore, no questionable ‘prelimiting’ of antidiffusive fluxes is required, which eliminates the danger of artificial steepening. Three FEM-FCT algorithms based on the Runge-Kutta, Crank-Nicolson, and backward Euler time-stepping are proposed. Numerical results are presented for the linear convection equation as well as for the shock tube problem of gas dynamics.