Universal Limiter for Transient Interpolation Modeling of the Advective Transport Equations : The ULTIMATE Conservative Difference Scheme

steps and shock waves, for example; but progress has been unexpectedly slow. It seems that correction of one defect always introduces another, equally severe. Unphysical oscillations inherent in classical second-order methods were eliminated by switching to first-order upwinding; but this merely replaced unacceptable oscillations with (what was ultimately realized to be) unacceptable global artificial diffusion. By devis- ing methods with locally varying artificial diffusion (small in smooth regions, larger in sharply varying regions), it is possible to achieve somewhat better resolution than global first-order upwinding without introducing spurious numerical oscillations. Some forms of shock-capturing (or TVD) schemes achieve their impressive results for step resolution by the use of locally varying positive artificial diffusion or viscosity (first-order upwinding) to suppress oscillations, com- bined with local negative viscosity (such as first-order downwinding) to arti- ficially compress or steepen the front. Unfortunately, this inherent negative diffusion is responsible for artificial steepening of (what should be) gentle gradients, as well, as will be demonstrated. Because of the concomitant flat- tening of local extrema (due to the local positive artificial diffusion), this defect has become known as "clipping," although the problem is initiated by the artificial steepening introduced to give high resolution to simulated fronts. In some cases (for example, a step-function followed