Constructing Polynomial Block Methods

TL;DR

This paper advances polynomial time integration by developing new polynomial block methods with imaginary nodes, improving their ability to solve dispersive equations and enabling more efficient serial computations.

Contribution

Introduces new polynomial block methods with imaginary nodes, avoiding algebraic order conditions and enhancing applicability to dispersive equations and serial computations.

Findings

New methods outperform backward difference methods.

Enhanced ability to solve dispersive equations.

Improved efficiency in serial computations.

Abstract

The recently introduced polynomial time integration framework proposes a novel way to construct time integrators for solving systems of first-order ordinary differential equation by using interpolating polynomials in the complex time plane. In this work we continue to develop the framework by introducing several additional types of polynomials and proposing a general class of construction strategies for polynomial block methods with imaginary nodes. The new construction strategies do not involve algebraic order conditions and are instead motivated by geometric arguments similar to those used for constructing traditional spatial finite differences. Moreover, the newly proposed methods address several shortcomings of previously introduced polynomial block methods including the ability to solve dispersive equations and the lack of efficient serial methods when parallelism cannot be used. To validate our new methods, we conduct two numerical experiments that compare the performance of polynomial block methods against backward difference methods and implicit Runge-Kutta schemes.