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

A second-order finite volume scheme is proposed and analyzed for a 2X2 system of non-linear partial differential equations. These equations model the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. The well-balancedness of the scheme is ensured through a modified limitation approach allowing 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. It is also shown through the numerical experiments that the second-order scheme reduces the 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 first-order schemes existing in the literature.