Doubly Monotonic Constraint on Interpolators: Bridging Second-Order to Singularity Preservation to Cancel “Numerical Wetting” in Transport Schemes

Abstract

Monotonic interpolation and its avatars are major ingredients of many numerical schemes for solving partial differential equations (PDEs) under total variation diminishing (TVD) or similar constraints. However, despite over forty years of extensive study of principles and applications, a key aspect of monotonic interpolant design can still appear somewhat empirical: how does a monotonic interpolator connect the limiting cases of smooth (differentiable) and singular (limited) functions in a consistent and possibly canonical way? The present study aims at providing understanding in the basic but important case of per-cell monotonic one-dimensional scalar reconstruction and at applying it to second-order accurate transport. First, a general mapping of bounded monotonic functions in elliptic coordinates is built. Then, the usual “single-slope” second-order monotonic interpolants are continued into “slope-and-bound” monotonic interpolants. Finally, a critical constraint is introduced, the “double monotonicity,” in order to build various slope-and-bound monotonic interpolators from this set of interpolants. With these slope-and-bound interpolators, standard numerical tests show a complete cancellation of the “numerical wetting” that usual TVD transport schemes produce. When transporting scalar fields of compact support, this effect---not to be confused with usual numerical diffusion---is the low-level contamination that spreads linearly in time over all the regions of the computational domain where nonvanishing transport is present. Removal of numerical wetting is of particular importance in many industrial and academic applications, notably at “phase disappearance” episodes in multiphase flows or “wet-dry” transitions in shallow water flows. Improvement of the “numerical erosion” of extrema is also observed. The general principles exposed here can be extended to multidimensional settings, high-order schemes, and other PDEs.