Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping methods for partial differential equations (PDEs) with stability characteristic of implicit methods. KSS methods compute each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE. As a result, KSS methods scale more effectively to higher spatial resolution than other time-stepping approaches.  Unfortunately, they are limited to linear PDEs.  This talk will present two avenues for broadening their applicability.

Exponential Rosenbrock methods are designed for stiff problems such as systems of ODEs that arise from spatial discretization of PDEs; however, these methods rely on computing matrix function-vector products with algorithms that do not scale well. KSS methods’ frequency-dependent approach, designed to circumvent stiffness in linear problems, computes these products with greater scalability. We demonstrate that combining these two classes of methods produces superior scalability for the solution of nonlinear problems. 

The talk will conclude with a presentation of explicit and implicit multistep formulations of KSS methods to provide a ``best-of-both-worlds’’ situation that combines the efficiency of multistep methods with the stability and scalability of KSS methods. The effectiveness of these ``spectral multistep methods’’ will be demonstrated through numerical experiments. It will also be shown that the region of absolute stability exhibits striking behavior that helps explain the scalability of these methods.

Joint work with Chelsea Drum and Bailey Rester.