Matrix Approximations Of Sturm-liouville Operators And Their Applications

Sturm-Liouville (SL) operator is one of the most important and ubiquitous differential operator in mathematics and physics. The spectral approximation of SL operator itself is very important in many practical applications, meanwhile it plays a fundamental role in many numerics of ODEs and PDEs. The highly oscillatory property of high order eigenfunctions makes the computation of a large number of eigenpairs a challenging task. Existing numerical methods are still not accurate and efficient enough. For example, the traditionally used finite difference and finite element methods usually discretize a SL problem into a generalized matrix eigenvalue problem, and only a small part of the eigenpairs of this matrix eigenvalue problem are accurate.An ideal approximation operator for a SL operator is the one with all its eigenpairs accurately approximating those of SL operator. The main purpose of this thesis is to find finite matrices to approximate SL operators as accurate as possible. From the viewpoint of function approximation, this is equivalent to finding a good subspace of the infinite solution space or its dual space. Then we can construct differentiation matrices to approximate differential operators by interpolation projection on this subspace. The approximation accuracy is com-pletely determined by interpolation bases, points and formulas. By considering the convergence, stability and computational efficiency of interpolation process, we systematically construct differentiation matrices based on both algebraic and trigonometric polynomials. Theoretical analysis and numerical results verify the high accuracy of these differentiation matrices for the approximation of SL op-erators.For algebraic polynomial bases, it is well-known Chebyshev points and barycen-tric formula are the near best choice. We introduce a cancelation treatment of boundary conditions, then every SL problem can be discretized into a standard (not generalized) matrix eigenvalue problem. We show Chebyshev differentiation matrix (CDM) is an ideal approximation for a class of singular SL operators. For regular SL operators, we generalize its 2/πconvergence property for constant coef-ficient SL problems to variable coefficient SL problems. We also use the mapped barycentric CDM (MBCDM) to solve SL problems with a higher accuracy and generalize its 2arcsinα/παconvergence property for constant coefficient SL problems to variable coefficient SL problems.For trigonometric polynomial bases, we use different bases for different boundary conditions so that the boundary conditions are automatically satis-fied. Besides the well-known Fourier differentiation matrix for periodic boundary conditions, we construct new barycentric interpolation formulas and new trigono-metric differentiation matrices for other boundary conditions. These trigonomet-ric differentiation matrices are particularly appropriate for the approximation of SL operator whose eigenfunctions oscillate uniformly.Finally, we also study several applications of the matrix approximations of SL operators. Specifically, the spectra computation of Schrodinger operators and the modal computation of waveguides; the numerical solution of Helmholtz equa-tion with high wavenumber; the numerical solution of time evolutionary PDEs and one-way Helmholtz equation by combining high accuracy differentiation ma-trix with exponential integrator. In particular, we study the artificial boundary conditions and PMLs and explain the pollution effect in the numerical solution of high wavenumber Helmholtz from the viewpoint of operator spectral approxi-mation.