Exponential Polynomial Block Methods

TL;DR

This paper introduces a new class of high-order, parallelizable exponential polynomial block methods for solving initial value problems, offering improved stability and efficiency over existing exponential integrators.

Contribution

The paper extends polynomial time integration to exponential methods and constructs a novel class of parallel exponential polynomial block methods based on Legendre points.

Findings

High-order EPBMs can be constructed with arbitrary accuracy.

EPBMs demonstrate improved stability over existing methods.

EPBMs are more efficient for parallelizable RHS evaluations.

Abstract

In this paper we extend the polynomial time integration framework to include exponential integration for both partitioned and unpartitioned initial value problems. We then demonstrate the utility of the exponential polynomial framework by constructing a new class of parallel exponential polynomial block methods (EPBMs) based on the Legendre points. These new integrators can be constructed at arbitrary orders of accuracy and have improved stability compared to existing exponential linear multistep methods. Moreover, if the ODE right-hand side evaluations can be parallelized efficiently, then high-order EPBMs are significantly more efficient at obtaining highly accurate solutions than exponential linear multistep methods and exponential spectral deferred correction methods of equivalent order.