A Single Stage Flux-Corrected Transport Algorithm for High-Order Finite-Volume Methods

TL;DR

This paper introduces a novel flux-corrected transport limiter for high-order finite-volume advection schemes, enhancing accuracy at smooth extrema and capturing discontinuities effectively with minimal limiting steps.

Contribution

It proposes a modified flux limiter with improved solution bounds and a single limiting application per time step, using advanced flux computation methods.

Findings

Retains high-order accuracy for smooth solutions

Accurately captures discontinuous fronts

Requires only one limiter application per time step

Abstract

We present a new limiter method for solving the advection equation using a high-order, finite-volume discretization. The limiter is based on the flux-corrected transport algorithm. We modify the classical algorithm by introducing a new computation for solution bounds at smooth extrema, as well as improving the pre-constraint on the high-order fluxes. We compute the high-order fluxes via a method of lines approach with fourth order Runge-Kutta as the time integrator. For computing low-order fluxes, we select the corner transport upwind method due to its improved stability over donor-cell upwind. Several spatial differencing schemes are investigated for the high-order flux computation, including centered difference and upwind schemes. We show that the upwind schemes perform well on account of the dissipation of high wavenumber components. The new limiter method retains high-order accuracy for smooth solutions and accurately captures fronts in discontinuous solutions. Further, we need only apply the limiter once per complete time step.