Krylov Subspace Spectral Methods for Systems of Variable-Coefficient PDE

For scalar time-dependent variable-coefficient PDE, it has been demonstrated that Krylov subspace spectral (KSS) methods achieve high-order accuracy and also possess highly desirable stability properties, especially considering that they are explicit. In this paper, we examine the generalization of these methods to systems of variable-coefficient PDE by selection of appropriate bases of trial and test functions. Furthermore, we show that for certain special cases, even higher-order accuracy is possible, as has previously been established in the case of the scalar second-order wave equation. Finally, we consider the use of KSS methods to obtain high-order operator splittings for systems of equations.