Shock capturing and front tracking methods for granular avalanches

TL;DR

This paper develops and tests shock-capturing numerical schemes combined with front-tracking methods to accurately simulate shock formations and free boundary dynamics in granular avalanches.

Contribution

It introduces a novel combination of NOC schemes with front-tracking for granular flow modeling, improving shock and boundary resolution.

Findings

NOC scheme with front-tracking accurately reproduces shocks and smooth solutions.

Piecewise quadratic WENO enhances solution smoothness near extrema.

Methods validated against exact solutions for various avalanche scenarios.

Abstract

Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shock-capturing numerical scheme for the one-dimensional Savage-Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the non-conservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A non-oscillatory central difference (NOC) scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a front-tracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the front-tracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane and (3) the motion of a parabolic cap down a curved slope ending in a flat run-out region, where a shock is formed as the avalanche comes to a halt.