High-resolution FEM–FCT schemes for multidimensional conservation laws

Abstract

The flux-corrected-transport paradigm is generalized to implicit finite element schemes and nonlinear systems of hyperbolic conservation laws. In the scalar case, a nonoscillatory low-order method of upwind type is derived by elimination of negative off-diagonal entries of the discrete transport operator. The difference between the discretizations of high and low order is decomposed into a sum of skew-symmetric antidiffusive fluxes. An iterative flux limiter independent of the time step is proposed for implicit schemes. The nonlinear antidiffusion is incorporated into the solution in the framework of a defect correction scheme preconditioned by the monotone low-order operator. In the case of a hyperbolic system, the global Jacobian matrix is assembled edge-by-edge without resorting to numerical integration. Its low-order counterpart is constructed by rendering all off-diagonal blocks positive definite or adding scalar artificial diffusion proportional to the spectral radius of the Roe matrix. The coupled equations are solved in a segregated manner within an outer defect correction loop equipped with a block-diagonal preconditioner. After a suitable synchronization, the correction factors evaluated for an arbitrary set of indicator variables are applied to the antidiffusive fluxes which are inserted into the global defect vector. The performance of the new algorithm is illustrated by numerical examples for scalar transport problems and the compressible Euler equations.