Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A well-balanced second-order finite volume approximation for a coupled system of granular flow

Aekta Aggarwal, Veerappa Gowda G. D., Sudarshan Kumar K

TL;DR

The paper tackles robust, high-fidelity numerical approximation of the Hadler–Kuttler sandpile model, a coupled non-linear hyperbolic system with a vertical source, on bounded domains. It develops a well-balanced, second-order finite-volume scheme by combining a MUSCL-type spatial reconstruction with SSP Runge–Kutta time stepping, and then introduces a novel adaptive limiter that reverts to first-order near equilibria to preserve the discrete steady state while maintaining second-order accuracy away from it. The resulting methods are analyzed for stability and well-balance in 1D and extended to 2D, with a comprehensive set of numerical experiments showing reduced oscillations, faster convergence to steady states, and sharper resolution of sandpile structures compared to first-order schemes. The approach, including the transport-ray based splitting of the source and the directional adaptive scheme, yields efficient and accurate simulations of granular flow under open and partially open boundary conditions, with potential applicability to broader discontinuous-flux hyperbolic systems.

Abstract

A well-balanced second-order finite volume scheme is proposed and analyzed for a 2 X 2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to reduce to well-balanced first-order scheme near the steady state while maintaining the second-order accuracy away from it. The well-balanced property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations, takes fewer time iterations for achieving the steady state and gives sharper resolutions of the physical structure of the sandpile, as compared to the existing first-order schemes of the literature.

A well-balanced second-order finite volume approximation for a coupled system of granular flow

TL;DR

The paper tackles robust, high-fidelity numerical approximation of the Hadler–Kuttler sandpile model, a coupled non-linear hyperbolic system with a vertical source, on bounded domains. It develops a well-balanced, second-order finite-volume scheme by combining a MUSCL-type spatial reconstruction with SSP Runge–Kutta time stepping, and then introduces a novel adaptive limiter that reverts to first-order near equilibria to preserve the discrete steady state while maintaining second-order accuracy away from it. The resulting methods are analyzed for stability and well-balance in 1D and extended to 2D, with a comprehensive set of numerical experiments showing reduced oscillations, faster convergence to steady states, and sharper resolution of sandpile structures compared to first-order schemes. The approach, including the transport-ray based splitting of the source and the directional adaptive scheme, yields efficient and accurate simulations of granular flow under open and partially open boundary conditions, with potential applicability to broader discontinuous-flux hyperbolic systems.

Abstract

A well-balanced second-order finite volume scheme is proposed and analyzed for a 2 X 2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to reduce to well-balanced first-order scheme near the steady state while maintaining the second-order accuracy away from it. The well-balanced property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations, takes fewer time iterations for achieving the steady state and gives sharper resolutions of the physical structure of the sandpile, as compared to the existing first-order schemes of the literature.
Paper Structure (17 sections, 3 theorems, 122 equations, 12 figures, 3 tables)

This paper contains 17 sections, 3 theorems, 122 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Numerical schemes in one-dimension
  3. First-order scheme
  4. Second-order scheme
  5. Boundary conditions for the second-order scheme
  6. Stability results in one-dimension
  7. Open table problem in one-dimension and well-balance property
  8. Second-order adaptive scheme in one-dimension
  9. Numerical schemes in two-dimensions
  10. Second-order scheme
  11. Second-order adaptive scheme
  12. Computation of the terms $B^x$ and $B^y$ in two dimensions
  13. Boundary conditions
  14. Numerical experiments
  15. Examples in one-dimension (1D)
  16. ...and 2 more sections

Key Result

Theorem 1

Let $f\ge0$ in $\Omega$. Assume that ${ \sup_{i\in \mathbb{Z}} |\alpha_i^0|\leq1},$${u_{i+\frac{1}{2}}^0}\ge0$ and $v_i^{0}\ge0$, for all $i\in \mathbb{Z},$ then the numerical scheme (rkfalpha) under the CFL conditions: satisfies the following properties for all $i \in \mathbb{Z}$:

Figures (12)

  • Figure 1: An element $C_{i,k}$ of the two dimensional Cartesian grids
  • Figure 2: The domain $\Omega$ and the boundary $\Gamma$ with transport rays.
  • Figure 3: Example 1 (1D): numerical solutions computed at time $t = 1.3.$ with $\Delta x = 1/100,\, \Delta t = 0.45 \Delta x.$
  • Figure 4: Example 3 (1D): numerical solutions near the steady state, computed at time $T=450$ with $\Delta x = 1/100, \Delta t = 0.45 \Delta x.$ (b) and (d) are enlarged view of (a) and (c), respectively.
  • Figure 5: Example 4 (1D): numerical errors versus number of time iterations plots. Numerical solutions are computed till it reach near the state with $\Delta x = 1/100, \Delta t = 0.45h.$ In each iteration, the errors produced by FO, SO and SO-$\Theta$ schemes are compared.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3