中文摘要：

非线性Poisson方程在化学、化工及生物等领域有着广泛的应用。本文发展了一种基于格子演化的新算法一格子Poisson方法（LPM），并且给出了Dirichlet边界条件和Neumann边界条件的实现方法。本方法不需要对方程进行线化处理，直接求解非线性方程，适用范围广泛。Dirichlet边界与Neumann边界的数值模拟结果与多重网格法等结果符合很好，验证了该方法在求解非线性Poisson方程的正确性与有效性。本方法非常适合并行计算，并方便扩展到三维情况。

英文摘要：

The nonlinear Poisson equation play a fundamental role in many biochemical and biophysical processes, such as bio-macromolecules interactions in electrolyte solutions. This paper presents a new lattice evolution based algorithm, Lattice Poisson method （LPM）, for solving the nonlinear Poisson equation without any linearization process. The high-order accurate boundary implements involving Dirichlet and Neumann boundaries are proposed in details. The LPM results agree well with the classical partial difference equation solutions, such as the multigrid solutions, for various cases with Dirichlet or Neumann boundary conditions. The present lattice evolution based method is suitable for parallel computing and can be easily extended to three dimensional cases.