Robust Numerical Methods For Problems With Boundary/Interior Layers

Various types of boundary/interior layers appear in the mathematical physics problems of science & technology and engineering,for example,boundary layers in viscous incompressible flow,shock layers in compressible gas dynamics,initial layers in some hyperbolic equations,etc.The most important feature for this kind of problem is that,in a very narrow region,the solution of the equation or its space/time derivatives will change rapidly.Traditional numerical methods are prohibitively expensive for this kind of problem.Therefore,other techniques are needed to design more efficient numerical methods.In this thesis,we apply the asymptotic analysis and other applied mathematical tools to the numerical solutions for problems with boundary/interior layers in space and time,and robust numerical methods are obtained.For problems with boundary/interior layers in space,we study the Oseen flow with large Reynolds number,the heterogenous radiation diffusion problem,and the singularly perturbed nonlinear eigenvalue problem arising in Bose-Einstein condensation.The Oseen flow is the linearization of the incompressible Navier-Stokes flow.By the equation decomposition technique and the artificial boundary method,the Oseen flow problem can be transferred into an equivalent second order elliptic boundary value problem on a bounded computational domain.Then we can solve the auxiliary problem by the tailored finite point method.Numerical results show that our method is stable w.r.t.Reynolds number,and we can capture the boundary/interior layers even on coarse mesh.The nonequilibrium radiation diffusion equations depict inertial confinement fusion process,and we propose the monotone finite point method for the numerical simulation.Our method can preserve the properties of monotonicity and positivity.Numerical results show that our method can capture the sharp front and can be accommodated to discontinues diffusion coefficient.The singularly perturbed nonlinear eigenvalue problem in this thesis depicts Bose-Einstein condensation in the strong repulsive regime.Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions.Combined with the normalized gradient flow method,we propose the adaptive finite element method with hybrid basis,in which the basis functions are chosen adaptively to the mesh partition and the asymptotic solution.For a simpler singularly perturbed two-point boundary value problem,we can prove our method to be uniformly convergent.For problems with initial layers in time,we study the singularly perturbed Kuramoto-Sivashinsky equation.This wave equation depicts the flame front propagation in a two dimensional free interface problem.Multiple time scales will appear in the singularly perturbed case.More precisely,besides the asymptotic solution of the limiting Kuramoto-Sivashinsky equation,the solution of the wave equation will have an additional initial layer,leading to a solution with large value for the second order(or higher order)derivative in time.We propose two kinds of numerical methods,the implicit-explicit Fourier pseudospectral method and the exponential wave integrator Fourier pseudospectral method,and prove them to be uniformly convergent.In particular,the latter is proved to be uniformly and optimally second order convergent in time for well-prepared initial data.