The Study On Numerical Methods For Stochastic Partial Differential Equations Based On Monte Carlo Method

In this paper,we study the partial differential equations with random coefficients.Since the birth and development of computer,researchers have made extensive and in-depth research on the intricate problems described by partial differential equations in the past.The partial differential equations include deterministic partial differential equations(PDE)and stochastic partial differential equations(SPDE).In other words,when studying a problem,if only the main influencing factors are considered,the mathematical model established for it is generally certain.In order to study the problem more deeply and specifically,some seemingly small influencing factors should also be considered,especially some important random factors.Stochastic partial differential equation,which is a class of equations used to calculate how complex systems evolve when random factors have to be taken into account.It has been used widely in a number of fields such as thermal science,geology,physics,chemistry,optics and so on.It also plays a significant part in demography,economy,finance.At present,it has become a popular research spot in probability theory research.However,due to the influence of random factors,the past numerical methods for solving deterministic problems are no longer applicable.For this purpose,a numerical method based on Monte Carlo method combined with finite difference method is studied for stochastic partial differential equation(group)problems.The research of stochastic partial differential equation is mainly divided into two parts:In the first part,based on the analysis of the heat conduction process of double glazing,the heat conduction problem and the optimal thickness of the gas interlayer between double glazing are considered in the multi-media coupling model.Firstly,a one-dimensional heat conduction equation is established based on thermal conductivity analysis.Due to technological problems,the gas density of the gas interlayer is a random variable,so the whole model is a random partial differential equation.The inner boundary coupling conditions,the first kind of left boundary conditions and the third kind of right boundary conditions are given according to the equal heat flux.Because of the randomness of the coefficients of the model equation,this paper uses Monte Carlo method and finite difference method to carry out random numerical simulation.Spatially,the difference implicit scheme is constructed and its truncation error and stability are analyzed.The random coefficients are randomly sampled and brought into the model to solve the corresponding "deterministic" problem.The optimum thickness of gas interlayer is obtained under given conditions.In the second part,this paper proposes a two-grid algorithm based on Monte Carlo modified finite difference scheme of windward block center in order to solve the problem of stochastic convection-diffusion response equation.To obtain the high precision numerical scheme,the convective term is modified and the block center finite difference method is used to construct the numerical scheme,so that the concentration scalar flux and vector flux can maintain the second order precision with respect to the spatial mesh size.Due to the randomness of the convective coefficient of the equation,this paper uses Monte Carlo random sampling to generate random coefficient field into the model for solving,and takes the mean value of the solution as the numerical solution of the model under the condition of fixed solution.In addition,in order to solve the semilinear convection-dominated problem effectively,a double grid algorithm is constructed in this paper.Finally,several numerical examples are given to verify the effectiveness and robustness of the algorithm in solving the stochastic convection-diffusion response problem.