The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection

Abstract

Modeling of highly advective transport is embarrassingly difficult, even in the superficially simple case of one-dimensional constant-velocity flow. In this paper, a fresh approach is taken, based on an explicit conservative control-volume formulation that makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This ‘ULTIMATE’ conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three representative test profiles: a discontinuous step, an isolated sine-squared wave, and a semi-ellipse. The goal is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservative advection schemes of any order of accuracy. Second-order methods are shown to be unsatisfactory because of steepening and clipping typical of currently popular so-called ‘high resolution’ shock-capturing or TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. By using adaptive stencil expansion to introduce (in principle arbitrarily) higher order resolution locally in isolated regions of high curvature or high gradient, extremely accurate coarse-grid results can be obtained with very little additional cost above that of the base (third-order) scheme.