High-order linearly implicit schemes conserving quadratic invariants

TL;DR

This paper introduces high-order, linearly implicit numerical schemes that preserve quadratic invariants in ordinary differential equations, combining efficiency and accuracy for physical and computational applications.

Contribution

It develops a unified method based on canonical Runge--Kutta methods to construct high-order, linearly implicit conservative schemes for quadratic invariants, filling a gap in existing methods.

Findings

Schemes preserve quadratic invariants exactly.

Constructed schemes are high-order and linearly implicit.

Proved properties related to accuracy and conservation.

Abstract

In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for preserving them have been extensively studied. Since linear invariants can be easily kept after discretisation, quadratic invariants are essentially the simplest ones. Quadratic invariants are important objects that appear not only in many physical examples but also in the computationally efficient conservative schemes for general invariants such as scalar auxiliary variable approach, which have been studied in recent years. It is known that quadratic invariants can be kept relatively easily compared to general invariants, and indeed can be preserved by canonical Runge--Kutta methods. However, there is no unified method for constructing linearly implicit and high order conservative schemes. In this paper, we construct such schemes based on canonical Runge--Kutta methods and prove some properties involving accuracy.