In this paper, we design and analyze second order positive and free energy satisfying schemes for solving diffusion equations with interaction potentials. The semi-discrete scheme is shown to conserve mass, preserve solution positivity, and satisfy a discrete free energy dissipation law for nonuniform meshes. These properties for the fully-discrete scheme (first order in time) remain preserved without a strict restriction on time steps. For the fully second order (in both time and space) scheme, a local scaling limiter is introduced to restore solution positivity when necessary. It is proved that such limiter does not destroy the second order accuracy. In addition, these schemes are easy to implement, and efficient in simulations. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes.

在本文中，我们设计和分析了二阶正和自由能满足方案，用于求解具有相互作用势的扩散方程。结果表明，半离散方案可以节省质量，保持溶液的正性，并满足非均匀网格的离散自由能耗散定律。完全离散方案（时间上的一阶）的这些属性得以保留，而对时间步长没有严格的限制。对于完全二阶（在时间和空间上）方案，需要时引入局部缩放限制器以恢复解的正性。事实证明，这种限幅器不会破坏二阶精度。此外，这些方案易于实现，并且在仿真中效率很高。给出了一维和二维数值示例，以演示这些方案的性能。