Implicitly Defined High-Order Operator Splittings for Parabolic and Hyperbolic Variable-Coefficient PDE Using Modified Moments

James Lambers
This paper presents a reformulation of Krylov Subspace Spectral (KSS) Methods, which use Gaussian quadrature in the spectral domain compute high-order accurate approximate solutions to variable-coefficient parabolic and hyperbolic PDE. This reformulation serves two useful purposes. First, it improves the numerical stability of these methods by removing cancellation arising from the approximation of certain derivatives by finite differences by computing these derivatives analytically. Second, it
more » ... reveals that KSS methods are actually high-order operator splittings that are defined implicitly, in terms of derivatives of the nodes and weights of Gaussian quadrature rules with respect to a parameter. Efficient algorithms for computing these derivatives are provided, as well as the first application of KSS methods to systems of coupled PDE.