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.

翻译：针对由2X2非线性偏微分方程组描述的颗粒流动动力学问题，本文提出并分析了一种二阶有限体积格式。该方程组模拟了多维度中平板有界矩形区域上垂直源作用下沙堆动态生长过程。通过改进的限制性方法确保格式的平衡性，该方法使格式在稳态附近退化为平衡一阶格式，同时在非稳态区域保持二阶精度。从解析角度证明了一维情形下的平衡特性，并通过二维数值实验进行验证。数值实验表明：与传统一阶格式相比，该二阶格式能有效减少有限时间振荡，以更少时间步数达到稳态，并更清晰解析沙堆物理结构的锐利边界。