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Godunov-Type Numerical Methods for a Model of Granular Flow on Open Tables with Walls

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems

Published online by Cambridge University Press:  05 October 2016

Adimurthi ,
Aekta Aggarwal  and
G. D. Veerappa Gowda
Adimurthi*
Affiliation:
TIFR Centre for Applicable Mathematics, Bangalore 560065, India
Aekta Aggarwal*
Affiliation:
The Indian Institute of Management Indore, Prabandh Shikhar, Rau Pithampur Road, Indore, Madhya Pradesh 453556, India
G. D. Veerappa Gowda*
Affiliation:
TIFR Centre for Applicable Mathematics, Bangalore 560065, India
*
*Corresponding author. Email addresses:aditi@math.tifrbng.res.in (Adimurthi), aektaaggarwal@iimidr.ac.in (A. Aggarwal), gowda@math.tifrbng.res.in (G. D. V. Gowda)
*Corresponding author. Email addresses:aditi@math.tifrbng.res.in (Adimurthi), aektaaggarwal@iimidr.ac.in (A. Aggarwal), gowda@math.tifrbng.res.in (G. D. V. Gowda)
*Corresponding author. Email addresses:aditi@math.tifrbng.res.in (Adimurthi), aektaaggarwal@iimidr.ac.in (A. Aggarwal), gowda@math.tifrbng.res.in (G. D. V. Gowda)
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Abstract

We propose and analyse finite volume Godunov type methods based on discontinuous flux for a 2×2 system of non-linear partial differential equations proposed by Hadeler and Kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table. The problem considered here is the so-called partially open table problem where sand is blocked by a wall (of infinite height) on some part of the boundary of the table. The novelty here is the corresponding modification of boundary conditions for the standing and the rolling layers and generalization of the techniques of the well-balancedness proposed in [1]. Presence of walls may lead to unbounded or discontinuous surface flow density at equilibrium resulting in solutions with singularities propagating from the extreme points of the walls. A scheme has been proposed to approximate efficiently the Hamiltonians with the coefficients which can be unbounded and discontinuous. Numerical experiments are presented to illustrate that the proposed schemes detect these singularities in the equilibrium solutions efficiently and comparisons are made with the previously studied finite difference and Semi-Lagrangian approaches by Finzi Vita et al.

Keywords

Balance lawsdiscontinuous fluxgranular matterwell-balanced schemesfinite volume schemesfinite difference schemestransport rayswalls

MSC classification

Secondary: 35L65: Conservation laws 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 4 , October 2016 , pp. 1071 - 1105
Copyright
Copyright © Global-Science Press 2016 

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