The Study Of Discontinuous Galerkin Methods And Reduced Basis Methods For Stochastic Partial Differential Equations

In this paper,we study the numerical analysis of discontinuous Galerkin(DG)methods for solving partial differential equations,and applications in magnetohydro?dynamics(MHD),and the reduced basis method(RBM)for solving stochastic partial differential equations.The paper is mainly divided into two parts.The first part includes the numerical analysis of several types of DG method and the numerical simulation of MHD.For the numerical analysis,We mainly use a shift-ing technique to design a special projection and carefully analyze the boundedness of the special projection yielding optimal error estimates.We prove the optimal error esti-mates of the semidiscrete central DG method on the overlapping meshes for solving lin-ear hyperbolic equations.For the one dimensional case,we prove the optimal(k+1)-th order accuracy on the uniform mesh when the piecewise polynomials of degree at most k,pk,are used.We then extend the analysis to multidicensions on uniform Carte-sian meshes when piecewise tensor product polynomials,Qk,are used on overlapping meshes.The projection can help us to deal with the troublesome intercell terms in the approximate error.The numerical examples verify our theoretical results.We continue to use this shifting technique to study the optimal error estimates of the classical semidis-crete DG scheme for solving two-dimensional scalar hyperbolic equations on Cartesian meshes using Pk elements.We separately provide the optimal error estimates for three cases namely,linear constant coefficients,linear variable coefficients,and non-linear cases.In addition,we study the superconvergence properties of a new optimal energy-conserving DG method for one-dimensional linear hyperbolic equations.We use the correction function technique to prove that the numerical solution of the semidiscrete scheme has(2k+1)-th order convergence rates for the numerical fluxes and the cell averages.Furthermore,we prove the approximate solution superconverges to a partic-ular projection of the exact solution.The order of this superconvergence is proved to be k+2.We also find that the derivative and function value approximations of the DG solution are superconvergent at a class speical points,with an order k+1 and k+2,respectively.In addition,we study the numerical simulation of DG methods for the compressible MHD equations.We focus on two different important physical properties to develop two numerical schemes,the entropy stable nodal DG scheme on Cartesian coordinates and the locally divergence-free DG scheme on cylindrical coordinates.On Cartesian coordinates,we consider the symmetrizable MHD equations introduced by Godunov and study the entropy stability of the semidiscrete scheme on structured meshes.By us-ing suitable quadrature rules,entropy conservative fluxes within elements and entropy stable fluxes at elements interfaces,we obtain the entropy stability of the numerical scheme.For another import property of the MHD equations,the divergence-free condi-tion of the magnetic field,we design the locally divergence-free spectral-DG methods on cylindrical coordinates(r,?,z)for 3D MHD equations.For some special physical problems,we use the Fourier spectral method in the ?-direction and the DG approxi-mation in the(r,z)plane.By a careful design of the locally divergence-free set for the magnetic field,our spectral-DG methods are divergence-free inside each element for the magnetic field.Numerical examples are provided to demonstrate the efficiency and good performance of our proposed methods.The second part is about the reduced basis method for stochastic differential equa-tions.We propose,analyze and implement a new RBM tailored for the linear(ordinary and partial)differential equations(ODEs and PDEs)driven by arbitrary(i.e.not nec-essarily Gaussian)types of noise.There are four main ingredients in our algorithm.First,we propose a new space-time-like treatment of time in the numerical schemes for ODEs and PDEs that is based on time-stepping.The second ingredient is an accurate yet efficient compression technique for the spatial component of the space-time snap-shots that the RBM is adopting as bases.The third ingredient is a non-conventional"parameterization" of a non-parametric problem.The last is an RBM that is free of any dedicated offline procedure yet is still efficient online to deal with the resulting para-metric problem.The numerical experiments verify the effectiveness and robustness of our algorithms.