Uncertainty Quantification Methods For Several Types Of Stochastic Partial Differential Equations

With the rapid development of science and technology,scientific computing has become an important tool in many fields.Especially for some complex phys-ical problems,the experimental research methods are often expensive and difficult to repeat.Numerical simulation has become a significant means of scientific re-search.However,the uncertainty of parameters in the mathematical model of many problems universally exists.In order to calculate stochastic differential equations with uncertainty more accurately,it is necessary to design highly accurate numerical methods and preserve their convergence.In recent years,the uncertainty quantifi-cation method(UQ method)has become very popular and important,and has been used to solve stochastic differential equations by many researchers.The stochastic collocation method based on generalized polynomials chaos(gPC-SC method)and the stochastic Galerkin method based on the spectral decomposi-tion of generalized polynomials(gPC-SG method)are two important methods in the UQ framework.The main idea of the former method is as follows:Firstly,we take the random variables as the zeros of Askey orthogonal polynomials in the random space,then the stochastic differential equation turns to multiple deterministic dif-ferential equations at the zeros.Secondly,we compute the solutions of multiple deterministic differential equation.Finally,The numerical solution of stochastic d-ifferential equation is obtained by Lagrange interpolation.On the other hand,the main idea of the latter is as follows:Firstly,based on generalized polynomials chaos,we implement spectral decomposition on the solution of stochastic differen-tial equation in random space.Secondly,implementing Galerkin projection on its subspace,a set of equations about spectral decomposition coefficients are obtained.Finally,the numerical solutions are obtained by solving these equations.Inspired by this,the paper mainly discusses the above two UQ methods for three kinds of stochastic differential equations with random parameters:Maxwell equation with random parameters,nonlocal elliptic equation with random parameters and non-linear parabolic equation with random parameters.The content of the paper is as follows:For stochastic Maxwell equations with random parameters,the numerical scheme of gPC-SC method is obtained by using the central difference scheme in physical space-time and the Lagrange interpolation method in random space.The numerical scheme of gPC-SG method is obtained by using the Yee scheme in physical space-time and the spectral Galerkin method in random space.We prove that when the initial condition satisfies the k-order regularity condition,the solution of the equa-tion also has the same as the initial condition.Furthermore,we give the convergence analysis of the stochastic collocation method and the stochastic Garlerkin method.Finally some numerical examples are presented to support the analysis.When we use stochastic collocation method to study nonlocal elliptic equation-s with random parameters,spectral method is used in physical space,and Lagrange interpolation is used in random space.For gPC-SG method,we use biorthogonal polynomial technology to solve numerical solutions,that is,the numerical scheme of gPC-SG method is obtained by using the spectral Galerkin method both in the physical space-time and random space.The regularity analysis for the exact solu-tion and strict error estimate are carried out for the above methods.The theoretical predication is verified by numerical examples in the case of Hk and infinite smooth-ness for the solution.When we study the nonlinear Burgers equation and Allen-Cahn equation with random parameters,the numerical schemes of gPC-SC method are obtained by us-ing the spectral method in physical space,the Crank-Nicolson difference scheme in time,and the Lagrange interpolation method in random space.The regularity anal-ysis for the exact solution and strict error estimate are proved for the above method.We also use the cases of Hk and infinitely smoothness to test the theoretical results.