Application of the Entropy Viscosity Method and the Flux-Corrected Transport Algorithm to Scalar Transport Equations and the Shallow Water Equations

Abstract

The Flux-Corrected Transport (FCT) algorithm, in conjunction with the entropy viscosity method, was applied with the continuous finite element method to (a) a scalar transport equation that includes reaction and source terms; and (b) the shallow water equations. The resulting scheme shows convergence to the entropy solution, is positivity-preserving, and reduces or eliminates the onset of spurious oscillations. For smooth problems, second-order spatial accuracy is achieved if adequate solution bounds are used in the FCT algorithm. For the scalar transport equation, the method of characteristics is used to derive local solution bounds to impose on the numerical solution. For the shallow water equations, local transformations are made to characteristic variables for the limitation process of FCT. Explicit SSPRK time discretizations are considered for both scalar transport and the shallow water equations, and additionally for scalar transport, Theta time discretizations and steady-state are considered. Explicit FCT schemes are shown to be relatively robust. However, implicit/steady-state FCT schemes are shown to have significant nonlinear convergence issues in many cases.