抛物型随机偏微分方程的数值计算方法

Stochastic partial differential equations can describe the complex physical phenomena and reveal its important development variation. In many cases, the model established by stochastic partial differential equations is more accurate and objective than the model established by deterministic differential equations. The numerical solution of stochastic partial differential equations can be used to explain physical phenomena more precisely. The regularity of the solution to stochastic parabolic partial differential equations is rather weak. To get accurate numerical solution requires a large number of samples. Thus, the numerical solution is difficult to be obtained. The purpose of this project intends to study numerical methods for stochastic parabolic partial differential equations. There are two objects in this project. (1) Constructing the finite element method to solve the equation with stochastic coefficients and the right hand side of the equations containing random item; (2) Constructing the stochastic collocation method to solve the stochastic heat equations with the random boundary condition. The convergence and error analysis will also be discussed. The numerical method of this project will enrich the numerical tools for stochastic parabolic partial differential equations.

随机偏微分方程是描述复杂物理现象、揭示其发展变化规律的重要工具之一。在许多情况下，随机偏微分方程所建立的模型比确定性微分方程所建立的模型能够更准确、更客观地反映所描述现象的本质。随机偏微分方程的数值解可以合理地解释方程所描述的物理现象，有助于人们对物理现象有进一步的认识和分析。抛物型随机偏微分方程解的正则性很弱，同时为了得到高精度的数值解需要非常大的样本数目，所以数值求解难度非常大。本项目拟研究抛物型随机偏微分方程的数值计算方法。主要包括：(1)构造求解方程系数和右端项同时有随机项的抛物型随机偏微分方程的有限元方法并进行误差分析；(2)构造求解具有随机边界条件的热传导随机偏微分方程的配置法并进行误差分析。这些数值计算方法的研究将丰富抛物型随机偏微分方程的数值计算工具，在实际应用中有着重要的实用价值和现实意义。

本项目主要研究了抛物型随机偏微分方程的数值计算方法。主要针对具有实际应用背景的随机能源系统和随机价格系统开展数值计算方法的研究。通过研究系数间断随机微分方程的Heun法，构造了求解系数和右端项同时有随机项的抛物型随机偏微分方程的计算方法并进行了收敛性的讨论。针对具有随机边界条件的热传导随机微分方程构造了配置格式，并对配置法进行了误差分析。同时，本项目利用这些数值方法研究了随机能源系统和随机价格系统的稳定性与分岔等问题。这些数值计算方法的理论研究和实际应用，进一步加深了我们对一些实际物理现象的认识和分析。我们还研究了一类差分方程正解的存在性和一类半线性薛定谔泊松系统多包解的存在性。