Numerical Methods For Steady-state Singular Perturbational Problems

“Singular perturbation” means that a small perturbation may cause large rapid changes in solutions of mathematical and physical problems. The rapid changes often occur in very narrow regions which frequently adjacent to the boundaries of the domain because a small parameter multiplies the highest derivative in differential equations. Though the terminology “singular perturbation”was first used in 1955, the subject of numerical methods for singular perturbational problems is not a settled one and is still in a state of vigorous development.In this paper we are mainly focused on some special numerical methods to solve some steady-state singular perturbational problems, including global method, spline collocation method, compact difference method. The main research works are as follows:1. A theoretical convergence analysis is firstly given about the Galerkin method using Bernstein polynomials as basis for solving second-order differential equations. Then a least squares method and a collocation method based on the Bernstein polynomials are also proposed which can avoid numerical integrations.Illustrative examples including regularly and singularly perturbed ones are implemented to demonstrate the efficiency and the scope of application of these methods.2. Since global methods have limitation in solving singular perturbational problems with very thin boundary layer, we propose collocation method based on piecewise cubic Bernstein polynomials. The collocation method leads to a sparse algebra system with only five nonzero elements in each row of the coefficient matrix, it can be combined with uniform and non-uniform mesh conveniently. Numerical experiments with both uniform mesh and non-uniform mesh of Shishkin-type are presented to solve two-point boundary value problems.3. Piecewise Bernstein polynomials of arbitrary order are applied to solve two point boundary problems and the method can improve the computation accuracy a lot with the increasing of the order of the Bernstein polynomials.4. A new idea of direction–changing and order-reducing is proposed to generate a difference scheme over a five-point stencil for solving two-dimensional(2D)convection-diffusion equation with source term. The scheme is of positive type so it is unconditionally stable and the convergence rate is proved to be of secondorder. Fourth-order accuracy can be obtained by applying Richardson extrapolation algorithm. Numerical results show that the scheme is accurate, stable and especially suitable for convection-dominated problems with different kinds of boundary layers including elliptic and parabolic ones.The works in the thesis not only provide some new methods for solving singularly perturbed problems, the idea of these methods can also be applied to a wide variety of differential equations.