This present work is based on developing a deterministic discrete formulation for the approximation of a multidimensional Smoluchowski (Coalescence) equation on a nonuniform grid. The mathematical formulation of the proposed method is simpler, easy to implement, and focuses on conserving the first-order moment. The new scheme resolved the issue of mass conservation along individual components in contrast to the existing scheme which focuses only on conservation of the total mass of all components. The validation of the new scheme is conducted against the existing scheme by considering some classical tests. The comparison reveals that the new scheme has the ability to compute the higher-order moments with higher accuracy than the existing scheme on a coarse grid without taking any specific measures. For the higher-dimensional population balance equations, the mixing of components quantified using parameter is also computed accurately and efficiently using a very coarse nonuniform grid.

中文翻译：

---

非均匀网格上多维 Smoluchowski 方程的新有限体积方法

目前的工作基于为非均匀网格上的多维 Smoluchowski（聚结）方程的近似开发确定性离散公式。该方法的数学公式更简单，易于实现，并且侧重于一阶矩守恒。与仅关注所有组件总质量守恒的现有方案相比，新方案解决了单个组件的质量守恒问题。通过考虑一些经典测试，对新方案的验证是针对现有方案进行的。比较表明，在不采取任何具体措施的情况下，新方案能够在粗网格上以比现有方案更高的精度计算高阶矩。 还使用非常粗糙的非均匀网格准确有效地计算参数。