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Krylov subspace methods for variable-coefficient initial-boundary value problems

Posted on:2004-05-16Degree:Ph.DType:Thesis
University:Stanford UniversityCandidate:Lambers, James VincentFull Text:PDF
GTID:2450390011957163Subject:Mathematics
Abstract/Summary:
The design and analysis of numerical methods for the solution of partial differential equations of the form 6u6t (x, t) + L(x, D) u(x, t) = 0, where the differential operator L(x, D) has constant coefficients, is greatly simplified by the fact that, for many methods, a closed-form representation of the computed solution as a function of x and t is readily available. This is due in large part to the fact that for such methods, the matrix that represents a discretization of L(x, D ) is diagonalizable, and the eigenvalues and eigenfunctions of this matrix are known. For variable-coefficient problems, however, this simplification is not available.; This thesis presents an alternative approach to the solution of this problem in the variable-coefficient case that leads to a new numerical method, called a Krylov subspace method, for which the computed solution can easily be represented as a function of x and t. The method makes use of Gaussian quadrature in the spectral domain to compute Fourier components of the solution. For each component, a different approximation of the solution operator by a restriction to a low-dimensional Krylov subspace is employed, and each approximation is optimal in some sense for computing the given component. The computed solution can be analytically differentiated with respect to time, resulting in new approaches to deferred correction and the solution of PDE that are second-order in time such as the telegraph equation.; As the Krylov subspace method is more effective for problems where the operator L(x, D) has smooth coefficients, approaches to preconditioning differential operators using unitary similarity transformations are presented. These preconditioning techniques are based on the use of the uncertainty principle by Fefferman to obtain approximate diagonalizations of self-adjoint differential operators.
Keywords/Search Tags:Krylov subspace, Methods, Solution, Differential, Variable-coefficient
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