Spectral Methods For Some Partial Differential Equations

Numerical methods of partial differential equations are widely used in many fields,which benefit from the rapid development of computer technology and growth of application requirements,such as fluid mechanics,image processing,micro physics and so on.The main methods,including finite difference method,finite element method,finite volume method,spectral method etc.,take fully interested by researchers in many fields.In this dissertation,the spectral method is used to solve two fractional partial differential equations and a nonlinear Dirac system.First,this dissertation investigates a space-time fractional reaction-diffusion equation with viscosity terms.Furthermore,a space-time spectral Petrov-Galerkin method is established.The viscosity term is composed by the time and space Riemann-Liouville fractional derivatives.Owing to the global characteristic of spectral method and the fact that fractional derivative operator is nonlocal operator,spectral method is better used to solve fractional partial differential equations.The basis functions are established based on the generalized Jacobi functions with singular term because of the singularity of fractional derivatives.Compared with the approximation by standard orthogonal polynomials,it could reduce the computation cost by generalized Jacobi functions,which occurred in redundant computation made with the excessive approximation of singular terms by standard polynomials.Since the fractional operator is not a self-adjoint operator,the trial and test functions locate in different spaces.The spectral Petrov-Galerkin method is used to construct the trial and test functions by different basis functions,which makes more suitable to the problem.Based on the format of the basis functions corresponding to the trial and test functions,the existence and uniqueness of the weak solution of the problem is proved by Galerkin method.For the new basis functions,the global error estimate of the space-time spectral Petrov-Galerkin method is proposed.The spectral convergence order are presented in numerical results.What is more,the influence of viscosity term on diffusion is given by numerical solution based on the accuracy of the space-time spectral Petrov-Galerkin method.Second,a space-time spectral Petrov-Galerkin method is constructed to solve the third and fifth-order time fractional Kortewegde Vries-Burgers equations.Based on the generalized Jacobi functions and Legendre polynomials,the algorithm is constructed by the basis functions in both time and spatial directions,which keeps the singularity of the solution in time and ensures the smoothness of the solution in space.The numerical schemes by coefficient matrices,equivalent to the orignal schemes,are solved by Newton iteration method due to the nonlinear terms,and the numerical solutions are obtained by the matrices.According to the orthogonality of the basis function,the optimal error estiamtes of the numerical methods are proposed in weighted Sobolev space and the theoretical convergence order are proved.The convergence and its order are presented and verify the theoretical results by numerical results.At last,this dissertation proposes an efficient and accurate numerical method for a nonlinear Dirac system.For one-dimensional problems,the spectral Galerkin method is constructed by Hermite functions in the whole space,which reduces the error by approximating the unbounded domain to the bounded domain.The scalar auxiliary variable method is introduced to eliminate the conditional stability of the problem by the nonlinear term in the time direction.The unconditionally stable numerical iteration scheme is proposed.The nonlinear Dirac system satisfies the energy conservation property.Compared with the dissipative property of the backward difference formula of second-order,the Crank-Nicolson scheme would be more suitable to describe the actual physical process by numerical results in that it can ensure the conservation of discrete energy.The second-order convergence and unconditional stability of the numerical method are proved.The theoretical spectral convergence order in spatial direction and second-order convergence in time direction are verified in numerical results.Moreover,this dissertation performs the simulation of binary and ternary collisions.On the other hand,the numerical method can be applied to two-dimensional nonlinear Dirac system,which keeps the second-order convergence in time direction and spectral convergence in spatial direction.