Research on mathematical analysis of finite difference methods and its numerical simulations based on fictitious domain and distributions

1. 基于虚域和分布的有限差分方法的数值计算有限差分方法的数值计算公式是寻找在分布意义上满足偏微分方程的局部可积函数。这种想法自然导致了狄利克雷-诺伊曼映射在虚拟边界上的表达。证明了局部可积函数定义的数值解收敛于Schwartz分布意义下的真解。在此基础上，提出了一种利用快速傅里叶变换的数值算法。这是基于与傅里叶变换和泊松方程系数矩阵的特征值和特征向量的结合。从这个思想出发，我们构造了一种称为线路逐次消去的数值方法，并为传输问题的数值计算做出了贡献。应用数学问题及其数值模拟我们在这里发展的数值方法可以应用于环境数学中的各种问题。一是基于温度和大气的城市地表逆温的建模和模拟。我们展示了城镇奇异温度的数值模拟。二是研究油轮事故发生时溢油对海岸的污染。通过基于虚域法的数值模拟进行了验证。三是数值模拟在医疗中的应用。我们展示了脑脊液的数值模拟。如上所述，我们从虚拟域和分布的角度研究了有限差分方法的数学分析，并建立了适用于各种环境数学的数值算法。论文发表，研究成果在国际会议上发表。

1. Numerical calculation of finite difference methods based on fictitious domain and distributionsOur formulation for numerical calculation in finite difference method is to find the locally integrable function satisfying the partial differential equations in the sense of distribution. This idea leads naturally to the expression of Dirichlet-Neumann map on the fictitious boundary. We proved that numerical solutions defined by locally integrable functions are convergent to the genuine solution in the sense of the Schwartz distribution. Moreover we proposed new numerical algorithm which use the fast Fourier transform. This is based on the combination with the Fourier transform and the eigenvalues and eigenvectors of the coefficient matrix of Poisson's equation. From this idea we constructs a numerical method which we call the successive elimination of lines, and contribute to the numerical calculation for the transmission problem.2. Applied mathematical problem and its numerical simulationThe numerical method we developed here is possible to apply to various problems in environmental mathematics. One is the modeling and simulation for surface inversions which depend on the temperature and atmosphere in urban. And we showed the numerical simulation of the singular temperature in the town. Two is the research on the coastal pollution by spilled oil when it occurs in the case of the tanker accident. We showed it through the numerical simulation based on the fictitious domain method. Third is the numerical simulation in medical treatments. We showed a numerical simulation of the cerebrospinal fluid.As above mentiond, we researched mathematical analysis of finite difference methods from view points of fictitious domain and distribution, and established the numerical algorithm which is able to apply to various environmental mathematics. Our papers were published and research results were presented at international conferences.