引力透镜效应在天体物理领域有多方面重要应用。例如探测太阳系外三体系统中的冷行星并刻画其发生率，从而限制行星形成与演化理论；探测透镜星系中的低质量（$\lesssim 10^8 M_{\odot}$）子结构（暗物质或重子物质）的数目并刻画其分布，以检验冷暗物质模型。另外，在射电波段观测的引力透镜多重像还可能受电磁波传播效应影响，可据此探测透镜星系中的星际介质并刻画其性质，如电子密度、磁场强度等。以上应用均依赖于新的数值算法，本文围绕此开展了两方面研究。第一，三体微引力透镜光变曲线计算算法的开发及应用。当透镜系统含有三个天体时，其对背景源的引力聚焦放大率难以计算。此前普遍采用逆光线追踪法，需大量密集光线，算法时间和空间复杂度很高。本文实现了领域内首套三体透镜光变曲线开源计算工具，提出一种获取多重像的闭合连续边界的普适方法，基于轮廓积分法计算放大率，因此在背景源半径小的情况更高效。本文将其应用于 5 例事件的分析中，其中包括一颗可能位于双星系统中的类木星。此外，为详细分析各类三体系统的可探测性并预测巡天项目的产出，本文开发了一套三体微引力透镜事件模拟工具，并以此分析出给定一个类太阳-木星-土星系统，地球2.0卫星探测到该系统的概率约为$1\%$。还发现同一系统中多颗行星的可探测性会互相影响，存在不可忽略的抑制或促进作用。这表明在计算多行星系统的探测效率时，不能将各颗行星的探测效率简单相乘。第二，含射电波散射效应的强引力透镜系统模拟与建模方法。子结构可影响多重像的放大率并导致流量比异常现象，从而可被探测。甚长基线干涉测量技术在这方面起着重要作用。但星际介质会使射电波产生散射效应，也能导致流量比异常现象，不仅使建模更困难，还会与暗物质子结构这一成因发生简并。本文通过数值模拟探究了散射效应对多重像的影响，发现散射效应不仅造成像的展宽和流量密度降低，还可改变像的位置，这与观测现象相符。本文通过将引力透镜效应和散射效应分开处理，依次重建出透镜质量分布、背景源表面亮度分布、以及散射角分布模型。对于与 B0128+437 类似的系统，爱因斯坦角半径和散射角大小的重建误差分别为 $\lesssim 1\%$ 和 $\lesssim 10\%$。本文还发现对于单波段观测数据，使用透镜势修正可以解释由散射效应造成的流量密度比异常。这说明在使用这类系统进行与暗物质子结构相关的研究时，需谨慎判断该异常是否由传播效应导致。
Gravitational lensing has many important applications in astrophysics. For example, it can be used to detect cold exoplanets in triple systems and estimate their occurrence rate, to constrain planet formation and evolution theories. The detection of low mass ($\lesssim 10^8 M_{\odot}$) substructures (dark matter or baryonic matter) and the measurement of their abundance and distribution in lensing galaxies are important for testing the cold dark matter model. In addition, in the radio band, lensed images can also be affected by the propagation effects of electromagnetic wave caused by interstellar medium in the lensing galaxy. This effect can be used to measure the properties of interstellar medium, such as the electron density and magnetic field. These applications all rely on new numerical methods. This work carries out research in two related topics.The first is on the development and applications of light curve calculation algorithm for triple microlensing systems. When the lensing object contains three bodies, it is hard to calculate the lensing magnification for the background source. Traditional methods are based on inverse-ray tracing, which requires a large amount of densely covered light rays, and is time consuming. We developed the first open source tool in the community for the light curve calculations for triple microlensing systems. This tool is based on contour integration and includes a general method to obtain the closed continuous image boundaries of multiply lensed images. It is more efficient for small source radius than the inverse-ray tracing based methods. We have applied this tool to five triple microlensing events and found a Jovian planet possibly in a binary star system. In addition, to allow for detailed analyses of the detection properties of various triple-lens systems, as well as to predict the scientific outcomes of the upcoming microlensing surveys, we have developed a simulation tool for triple microlensing events. With this tool, we found that with the proposed Earth Two (ET) satellite, the detection probability of a Sun-Jupiter-Saturn analogue is about $1\%$, and that the detectability of multiple planets in the same system will affect each other. The suppression and enhancement effects are non-negligible, which means that it is inappropriate to treat each planet separately during detection efficiency calculations for multi-planet systems.The second aspect of this thesis is on the simulation and modelling methods for strong lensing systems which involve scattering effect of radio waves. Substructures can change the magnification of lensed images and cause flux-ratio anomaly, so that these substructures can be detected. The very long baseline interferometry (VLBI) technique plays an important role in detecting substructures. However, the interstellar medium can affect radio waves through various propagation effects, such as the scattering effect, which can also produce flux-ratio anomaly. The scattering effect not only complicates the modelling process, but also degenerate with substructures. With numerical simulations, we investigated the influence of the scattering effect on lensed images. We found that the scattering effect can change the size, flux density and position of the lensed image, which are consistent with observations. We demonstrated that by handling the lensing effect and the scattering effect separately, the lens mass distribution, the source surface brightness distribution, and the scattering angle distribution can be reconstructed. For a system similar to B0128+437, the errors on the reconstructed angular Einstein radius and the scattering angle are $\lesssim 1\%$ and $\lesssim 10\%$, respectively. In addition, we found that for single-band observation data, the potential correction can be used to explain the flux-ratio anomaly caused by the scattering effect. This means that when using similar lens systems to conduct dark matter substructure related studies, we need to carefully check whether the flux-ratio anomaly is caused by propagation effects.
