Eulerian-Lagrangian (EL-LG) scheme is a numerical scheme that tracks pseudo particles in Eulerian cells. It is widely used in the computational fluid dynamics, however, numerical errors associated with a particle source term has not yet been investigated much. Hence this study focuses on numerical errors of EL-LG caused by particle sources. The purposes are: (i) to clarify causes and situations that bring larger numerical errors by source terms and (ii) to suggest an idea to reduce them. For those purposes, we focus on the particle continuity equation and carry out systematic analysis of the numerical error by setting the following three simple cases: Case (A) No source, Case (B) Constant source, and Case (C) source with arbitrary spatial profile. For each case, we have obtained a theoretical expression of the numerical error. It has been clarified that the errors become relatively large when (i) the spatial profile of the particle source has a large gradient and (ii) the source is localized in the region with high flow-velocity. These were caused by the treatment of the particle source: If pseudo particles due to the source are added in a simple way at the start or the end of the time step, this can lead to larger numerical errors. To reduce those errors, a time-averaging scheme has been suggested. Although the analyzed cases are simple, the results obtained in this study would give important knowledge and insight into numerical errors associated with particle sources in EL-LG schemes.

欧拉-拉格朗日（EL-LG）方案是一种数值方案，用于跟踪欧拉细胞中的伪粒子。它广泛用于计算流体动力学，但是，与粒子源项相关的数值误差尚未得到足够的研究。因此，本研究关注由粒子源引起的EL-LG的数值误差。目的是：（i）通过源术语来澄清带来较大数值误差的原因和情况，以及（ii）提出减少误差的想法。为此，我们将重点放在粒子连续性方程上，并通过设置以下三个简单情况对数值误差进行系统分析：情况（A）无源，情况（B）恒定源和情况（C）任意的源空间轮廓。对于每种情况，我们都获得了数值误差的理论表达式。已经阐明，当（i）颗粒源的空间轮廓具有大的梯度并且（ii）源位于具有高流速的区域中时，误差变得相对较大。这些是由于对粒子源的处理引起的：如果在时间步骤的开始或结尾以简单的方式添加源于源的伪粒子，则可能导致更大的数值误差。为了减少这些错误，提出了一种时间平均方案。尽管分析的情况很简单，但本研究中获得的结果将为与EL-LG方案中与粒子源相关的数值误差提供重要的知识和见识。这些是由于对粒子源的处理引起的：如果在时间步骤的开始或结尾以简单的方式添加源于源的伪粒子，则可能导致更大的数值误差。为了减少这些错误，提出了一种时间平均方案。尽管分析的情况很简单，但本研究中获得的结果将为与EL-LG方案中与粒子源相关的数值误差提供重要的知识和见识。这些是由于对粒子源的处理引起的：如果在时间步骤的开始或结尾以简单的方式添加源于源的伪粒子，则可能导致更大的数值误差。为了减少这些错误，提出了一种时间平均方案。尽管分析的情况很简单，但本研究中获得的结果将为与EL-LG方案中与粒子源相关的数值误差提供重要的知识和见识。