Development of Reliability Evaluation System for Numerical Solutions by Introducing Stochastic Approach and Application to Complicated Fluid Dynamics Simulation

为了开发数值解的可靠性评估系统，本文从概率论的角度讨论了随机差分方程数值解的结构在随机误差插入后的变化.在求解非线性方程组时，由于误差的随机插入，使得解的结构变得非常复杂。作为基础研究，我们考虑了基于确定性Logistic微分方程和Lorenz方程的随机微分方程。本文提出了一种新的方法--样本平均动力系统（SMDS），用来分析离散动力系统数值解的结构对随机误差插入的依赖性，讨论了噪声大小与所得数值解的特性之间的关系，并应用SMDS方法对离散动力系统的数值解进行了分析 ...更多信息 艾德将他们引入到真实的流体计算问题和交通堵塞问题中，以分析影响非线性现象的几个因素。首先，我们试图讨论不可压缩流体方程数值解的结构对解联立方程时插入随机误差的依赖性。流体模拟的数值解的平均结构的依赖性，强行添加的随机误差进行了讨论。接下来，我们给出了一些理论上的考虑的通量自由有限元方法的广义Stokes界面问题所产生的不混溶的两流体流动问题。在无通量有限元法中，通量约束被设置为另一个拉格朗日乘子，以保持零通量的界面上。因此，预期每种流体的质量在每个时间步长处保持不变。我们非常仔细地快速研究了间断系数（粘性和密度）对标准有限元近似误差的影响。然后，将分析推广到无通量有限元法。针对交通拥挤问题，研究了双车道交通流中交通拥挤的形成。在单车道模型的基础上，引入微观模型中的最优速度模型，建立了双车道宏观模型。对每个车道采用不同的最优速度函数，并引入新的换道规则。为了研究双车道交通流的特征现象，我们进行了数值模拟，特别是我们集中讨论了“同步流”的性质，双车道交通流的最典型的现象之一，此外，从模拟的基本图表与在高速公路上观察到的比较。少

For the purpose of development of reliability evaluation system for numerical solutions, we tried to discuss how structure of numerical solutions of a stochastic difference equation changes by the insertion of errors with the random style from the viewpoints of probabilistic approaches. Errors in solving the nonlinear systems are inserted randomly and structure of solutions becomes very complicated We try to investigate the dependence of the structure of numerical solutions on insertion of random errors As a fundamental study, the stochastic differential equation based on the deterministic logistic differential equation and the Lorenz equations are considered. A new approach, sample mean dynamical system (SMDS), is proposed in order to analyze the dependence of the structure of numerical solutions of discretized dynamical system on insertion of random errors and the relation between the size of noise and characteristics of obtained numerical solutions is discussed.In addition, we appli … More ed them to the issues of real fluid calculation and the problem of traffic jam in order to analyze several factors which govern nonlinear phenomena. First, we tried to discuss the dependence of the structure of numerical solutions of incompressible fluid equations on insertion of random errors in solving simultaneous equations. Dependence of the averaged structure of numerical solutions of fluid simulations on forcibly added random errors are discussed. Next, we give some theoretical considerations on the flux-free finite-element method for the generalized Stokes interface problem arising from the immiscible two-fluid flow problems. In the flux-free finite-element method, the flux constraint is posed as another Lagrange multiplier to keep the zero-flux on the interface. As a result, the mass of each fluid is expected to be preserved at every time step. We fast study the effect of discontinuous coefficients(viscosity and density)on the error of the standard finite element approximations very carefully. Then, the analysis is extended to the flux-free finite element method. As for the problem of traffic jam, the formation of the traffic congestion in two-lane traffic flow is studied. The two-lane macroscopic model using the optimal velocity model which has been introduced in the microscopic model is constructed on the basis of the one-lane model. We adopt different optimal velocity function to each lane and new rules of changing lanes are introduced. Numerical simulations are performed in order to investigate the characteristic phenomena of two-lane traffic flow In particular, we concentrate the discussion about the property of "Synchronized flow', one of the most characteristic phenomena of two-lane traffic flow Furthermore, the fundamental diagrams from the simulations are compared with those observed in a highway. Less