Shock-capturing and front-tracking methods for granular avalanches

By Y. Tai, Sebastian Noelle, John Gray, Kolumban Hutter. 2015

159 citations recorded by the source corpus.

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 nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference 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.

Subfields: Computational Mechanics, Management, Monitoring, Policy and Law, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f8cae0ef7b2266daee5cb805a3be8bed7c4dbd24

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In Computational Mechanics

In Management, Monitoring, Policy and Law

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Citations in the corpus

Numerical modeling of avalanches based on Saint Venant equations using a kinetic scheme (2003, unknown): “Tai [2000] and Tai et al. [2002] overcome this imperfection by tracking the vacuum front.”
A general two-phase debris flow model (2012, methodology): “High-resolution, shock-capturing Total Variation Diminishing Non-Oscillatory Central (TVD-NOC) scheme is implemented to solve the model equations numerically [Nessyahu and Tadmor, 1990; Tai et al., 2002; Pudasaini et al., 2005; Pudasaini and Hutter, 2007; Pudasaini and Domnik, 2009] (Appendix D).”
Modelling debris flows down general channels (2005, background): “…Harten et al., 1986; Yee, 1987; Nessyahu and Tadmor, 1990; LeVeque, 1990; Kr̈oner, 1997; Jiang and Tadmor, 1998; Toro, 2001) and its application to avalanches is, e.g., given by Tai et al. (2002), Pudasaini (2003), Pudasaini and Hutter (2003a), Wang et al. (2004) and Pudasaini et al. (2005a, b).”
r.avaflow v1, an advanced open-source computational framework for the propagation and interaction of two-phase mass flows (2016, unknown): “The TVD-NOC scheme with the minmod limiter has successfully been applied to a large number of mass flow problems (Tai et al., 2002; Wang et al., 2004; Mergili et al., 2012; Pudasaini and Krautblatter, 2014; Kafle et al., 2016; Kattel et al., 2016).”
Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations (2001, unknown): “We consider Godunov-type schemes for the multidimensional (multi-D) systems of conservation laws ut +∇x · f(u) = 0, x ∈ Rd,(1.1) and the multi-D Hamilton–Jacobi equations ϕt +H(∇xϕ) = 0, x ∈ Rd.(1.2) Godunov-type schemes for the system (1.1) are projection-evolution methods.”
A new model of granular flows over general topography with erosion and deposition (2008, methodology): “, [37], [26]) or by utilizing Lagrangian moving grid methods (e.”