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.

翻译：针对描述多维平坦有界矩形桌上垂直源作用下增长沙堆动力学的2×2非线性偏微分方程组，提出并分析了一种二阶平衡有限体积格式。为推导二阶格式，我们将MUSCL型空间重构方法与强稳定保持龙格-库塔时间步进法相结合。通过改进的限制方法确保所得格式保持平衡性，该限制方法允许格式在稳态附近降阶为平衡一阶格式，同时在远离稳态时保持二阶精度。该格式的平衡特性在一维空间得到理论证明，并在二维空间进行数值验证。此外，数值实验表明，与现有的一阶格式相比，二阶格式能减少有限时间振荡，以更少的时间迭代达到稳态，并给出沙堆物理结构更清晰的解析结果。