Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws

In this paper, we focus on developing locally conservative high order finite difference methods with provable total variation stability for solving one-dimensional scalar conservation laws. We introduce a new criterion for designing high order finite difference schemes with provable total variation stability by measuring the total variation of an expanded vector. This expanded vector is created from grid values at tn+1 and tn with ordering determined by upwinding information. Achievable local bounds for grid values at tn+1 are obtained to provide a sufficient condition for the total variation of the expanded vector not to be greater than total variation of the initial data. We apply the Flux-Corrected Transport type of bound preserving flux limiters to ensure that numerical values at tn+1 are within these local bounds. When compared with traditional total variation bounded high order methods, the new method does not depend on mesh-related parameters. Numerical results are produced to demonstrate: the total variation of the numerical solution is always bounded; the order of accuracy is not sacrificed. When the total variation bounded flux limiting method is applied to a third order finite difference scheme, we show that the third order of accuracy is maintained from the local truncation error point of view.