Krylov Subspace Spectral Methods for Variable-Coefficient Initial-Boundary Value Problems
The design and analyis of numerical methods for the solution of PDE of the form du/dt + Lu = 0, where L is a constant-coefficient differential operator, is greatly simplified by the fact that, for many methods, a closed-form representation of the computed solution as a function of (x,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 is easily diagonalizable. For variable-coefficient problems, however, this simplification is not available. This talk describes an alternative approach to the variable-coefficient case that leads to a new numerical method, called a Krylov subspace spectral method, for which the computed solution can easily be represented as a function of (x,t). This function can be analytically differentiated with respect to t, resulting in new approaches to deferred correction and the solution of PDE that are second-order in time such as the telegraph equation. The basic idea is to use Gaussian quadrature in the spectral domain to compute components of the solution, rather than in the spatial domain as in traditional spectral methods. As the method is more accurate for problems where the operator L has smoother coefficients, approaches to preconditioning differential operators using unitary similarity transformations are also presented.