Efficient Numerical Methods For The Convection-diffusion Equation With Random Inputs

The convection-diffusion equation describes physical phenomena where par-ticles,energy,or other physical quantities are transferred inside a physical sys-tem.It is widely used in the fields of fluid mechanics,environmental science and energy development.However,in practical applications,some uncertainties often appear in the initial conditions,boundary conditions and parameters of the equation,making it become a convection-diffusion equation with random in-puts,i.e.,stochastic convection-diffusion equation.In order to solve a stochastic convection-diffusion equation,we need to discretize the probability space,the physical space and the time space respectively.How to design numerical meth-ods in the three types space to solve the stochastic convection-diffusion equation quickly and efficiently is the focal point of this thesis.The contents and results of this thesis are as follows:Part one,we study the time algorithm for solving stochastic convection-diffusion equations based on generalized polynomial chaos(gPC)method.By ap-plying the gPC method in the probability space,the original stochastic convection-diffusion equation is transformed into a set of deterministic coupled equations.Then how to efficiently solve the set of equations is the content of our study.The usual way to solve the set of equations is to use backward differentiation formula(BDF)in time,which is essentially a fully implicit scheme,but this scheme can-not decouple the equations.So we construct a implicit-explicit(IMEX)scheme which can decouple the equations.If the IMEX scheme is stable and feasible,it will improve the calculation efficiency and save time.We use the spectral col-location method in physical space.Numerical results show that the first-order IMEX,BDF1 and BDF2 schemes are stable and effective,and the second-order IMEX scheme is unstable.It is easily affected by the time stepperturbation parameters,and the highest order of the polynomial in the polynomial chaos expansion.Part two,we study the time algorithm for solving stochastic convection-diffusion equations based on stochastic collocation method.The key step of the stochastic collocation method is to first select Q special collocations in the probability space,and then solve the deterministic convection-diffusion equation corresponding to each collocation point.In general,a common way to solve each deterministic equation by using BDF1 and BDF2 schemes,which requires us to obtain Q different coefficient matrices and store the coefficient matrix Q times,and solve them one by one,which is inefficient.So we propose an improved time algorithm:ensemble time-stepping algorithm(ETSA).The proposed algorithm only need to solve a single linear system with one shared coefficient matrix,reducing both storge required and computational cost.We use the finite element method in physical space.The stability and error analysis of the first-and second-order ETSAs are provided.The numerical results are consistent with the theoretical analysis,and the first-and second-order ETSAs can achieve the same accuracy as BDF1 and BDF2 schemes,respectively.Part three,we study the radial basis function-generated finite difference(RBF-FD)method for solving convection-dominated diffusion problems.The convection-dominated diffusion problem is also a boundary layer or singular per-turbation problem.When some standard numerical methods are used to solve it,numerical oscillations often occur.So we consider a new hybrid discrete scheme base on a special piecewise equidistant(Shishkin)mesh for the problem.The hybrid scheme approximation to convection term with midpoint upwind scheme on the coarse mesh and standard central scheme on the fine mesh respectively.This scheme effectively improves the accuracy at the boundary layer and then improves the global accuracy.At the same time,the scheme is stable without numerical oscillations.The numerical results show that the RBF-FD method is more accurate than the finite difference method in the same Shishkin mesh with the same hybrid scheme.Part four,we study the two-level radial basis function method for solving the semilinear elliptic problem.For solving a nonlinear problem,a fast and effective method is to use two-level finite element method.Inspired by the two-level finite element method,we propose a two-level method based on radial basis functions.The main idea of this method is that in the first step,we use the RBF method to solve a semilinear problem with a small number of collocation points,in the second step,RBF-FD methods are used to solve the linearized problem with a large number of collocation points,respectively.Compared with the two-level finite element method,the method is more simple to operate.Numerical results show that the method is efficient and feasible.