Lattice Boltzmann Simulation Of Stochastic Burgers Equation

In many complicated systems, a great number of numerical simulations with uncertainties whichmay have insufficient knowledge of system, for example, the external forces, initial and boundaryconditions, or indeterminacy system parameters, become difficult to carry out. In most traditionalsystems especially having complex background, the practical phenomena are usually described bysome simplified mathematical models. Although these complex phenomena can be predicted by thesimplified models, they also have a lot of limitations because of the existence of some uncertainties.The lattice Boltzmann method, as a new numerical approach, not only has more advantages in thestudy of complex fluid flows, but also has a great potential in the simulation of nonlinear systems.Although lattice Boltzmann method has been widely used to solve many different types of partialdifferential equations, it receives less attention in solving stochastic partial differential equations. Inthis thesis we intend to extend the lattice Boltzmann method to study stochastic partial differentialequations.In this paper, we take the stochastic Burgers equation as an example. According to the randomitems in different parts, one and two-dimensional Burgers equations are divided into three cases:boundary perturbation, viscosity perturbation and source term perturbation. Each of the conditions wassimulated by the lattice Boltzmann method. For one-dimensional stochastic Burgers equation: theresults obtained by lattice Boltzmann simulation for boundary perturbation with uniform distributionare better than those derived by the generalized polynomial chaos expansion method; for boundary withthe Gaussian distribution, we constructed a new Gaussian distribution in a closed interval, and find thatour results have a good agreement with the existing literature; the numerical results for viscositycoefficient with the Gaussian distribution indicates that viscosity coefficient gives a significantinfluence on the waveform solution at steady time. For two-dimensional stochastic Burgers equation:the existing literature only analyze the supersensitivity for boundary perturbation case with severalfixed values, while in this thesis, we will further extend these available works, and study the boundarywith uniform and Gaussian distributions, which will be helpful to understand the solution sensitivitywith boundary perturbation from the statistical point of view; the numerical results of the Burgersequation with the Gaussian distribution source term show that the solution at steady state hassupersensitivity phenomenon to the perturbation of the source term.