In this paper, we propose a numerical method to uniformly handle the random genetic drift model for pure drift with or without natural selection and mutation. For pure drift and natural selection case, the Dirac $\delta$ singularity will develop at two boundary ends and the mass lumped at the two ends stands for the fixation probability. For the one-way mutation case, known as Muller's ratchet, the accumulation of deleterious mutations leads to the loss of the fittest gene, the Dirac $\delta$ singularity will spike only at one boundary end, which stands for the fixation of the deleterious gene and loss of the fittest one. For two-way mutation case, the singularity with negative power law may emerge near boundary points. We first rewrite the original model on the probability density function (PDF) to one with respect to the cumulative distribution function (CDF). Dirac $\delta$ singularity of the PDF becomes the discontinuity of the CDF. Then we establish a upwind scheme, which keeps the total probability, is positivity preserving and unconditionally stable. For pure drift, the scheme also keeps the conservation of expectation. It can catch the discontinuous jump of the CDF, then predicts accurately the fixation probability for pure drift with or without natural selection and one-way mutation. For two-way mutation case, it can catch the power law of the singularity. %Moreover, some artificial algorithms or additional boundary criteria is not needed in the numerical simulation. The numerical results show the effectiveness of the scheme.

翻译：本文提出一种数值方法，用于统一处理含或不含自然选择与突变的纯随机遗传漂变模型。在纯漂变与自然选择情形下，狄拉克$\delta$奇异性会出现在两个边界端点，两端聚集的质量代表固定概率。对于单向突变情形（即穆勒棘轮效应），有害突变的积累会导致最适基因的丢失，狄拉克$\delta$奇异性仅在一个边界端点处产生尖峰，这代表有害基因固定与最适基因丢失。对于双向突变情形，边界点附近可能出现负幂律奇异性。我们首先将原始概率密度函数（PDF）模型重新表述为关于累积分布函数（CDF）的形式。PDF的狄拉克$\delta$奇异性转化为CDF的不连续性。随后我们建立了一种迎风格式，该格式保持总概率守恒、满足正性保持且无条件稳定。对于纯漂变情形，该格式还保持期望守恒。它能捕捉CDF的不连续跳跃，从而精确预测含或不含自然选择的纯漂变及单向突变情形下的固定概率。对于双向突变情形，该格式能捕捉奇异性的幂律规律。数值结果验证了该格式的有效性。