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A Generalized Flux-Corrected Transport Algorithm I: A Finite-Difference Formulation

William J Rider, Dennis R Liles

TL;DR

Results show that the new FCT algorithm performs better than the older FCT algorithms and is comparable with other modern methods and will also allow the FCT to be used effectively with exact or approximate Riemann solvers and as an implicit algorithm.

Abstract

This paper presents a generalized flux-corrected transport (FCT) algorithm, which is shown to be total variation diminishing under some conditions. The new algorithm has improved properties from the standpoint of use and analysis. Results show that the new FCT algorithm performs better than the older FCT algorithms and is comparable with other modern methods. This reformulation will also allow the FCT to be used effectively with exact or approximate Riemann solvers and as an implicit algorithm. This paper was originally submitted to the Journal of Computational Physics in 1990. It got lost in review. One reviewer loved the paper and suggested it be published immediately (he also died while it was in review). Another reviewer savaged the paper being from the FCT camp. The journal also went through several changes in management. Ultimately I declined to continue pursuing the paper as I had one infant child at the time and another on the way in 1995. Now 30 years on I am going to put this online.

A Generalized Flux-Corrected Transport Algorithm I: A Finite-Difference Formulation

TL;DR

Results show that the new FCT algorithm performs better than the older FCT algorithms and is comparable with other modern methods and will also allow the FCT to be used effectively with exact or approximate Riemann solvers and as an implicit algorithm.

Abstract

This paper presents a generalized flux-corrected transport (FCT) algorithm, which is shown to be total variation diminishing under some conditions. The new algorithm has improved properties from the standpoint of use and analysis. Results show that the new FCT algorithm performs better than the older FCT algorithms and is comparable with other modern methods. This reformulation will also allow the FCT to be used effectively with exact or approximate Riemann solvers and as an implicit algorithm. This paper was originally submitted to the Journal of Computational Physics in 1990. It got lost in review. One reviewer loved the paper and suggested it be published immediately (he also died while it was in review). Another reviewer savaged the paper being from the FCT camp. The journal also went through several changes in management. Ultimately I declined to continue pursuing the paper as I had one infant child at the time and another on the way in 1995. Now 30 years on I am going to put this online.

Paper Structure

This paper contains 11 sections, 1 theorem, 99 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Method Development
  3. Zalesak's FCT Algorithm
  4. A New FCT Algorithm
  5. A Modified-Flux FCT Algorithm
  6. Extension of FCT to Systems of Equations
  7. Results
  8. Scalar Advection Equation
  9. Burger's Equation
  10. Sod's Shock Tube Problem
  11. Concluding Remarks

Key Result

Theorem 1

The new FCT algorithm without the monotone (TVD) first step is TVD under the following conditions: for $\tilde{\sigma} = \psi\left( a \right)$ and for $\tilde{\sigma} = \psi\left( a \right) -\lambda a^2$ Here $\theta$ is an implicitness parameter (see Yee yee87a) with $\theta = 1$ the equation is fully implicit, and with $\theta=0$ the equation is fully explicit. Without the monotone first step, t

Figures (18)

  • Figure 1: The characteristics of the FCT limiters for the modified-flux formulation.
  • Figure 2: Solution of the scalar advection equation with Zalesak's FCT with the high-order flux defined by second-order central differencing.
  • Figure 3: Solution of the scalar advection equation with the new FCT with the high-order flux defined by second-order central differencing.
  • Figure 4:
  • Figure 5: Solution of the scalar advection equation with the new FCT with the high-order flux defined by Lax-Wendroff differencing.
  • ...and 13 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3