Explicit invariant-preserving integration of differential equations using homogeneous projection

TL;DR

This paper introduces explicit invariant-preserving integrators for differential equations using homogeneous projection, enabling high-order, accurate, and efficient preservation of invariants in numerical solutions.

Contribution

It presents a novel homogeneous projection framework for explicit invariant-preserving integration, improving accuracy and efficiency over existing methods.

Findings

Methods preserve invariants with high accuracy.

Significant improvements in numerical accuracy and efficiency.

Applicable to complex nonlinear systems and PDEs.

Abstract

We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method exploits homogeneous symmetries to evaluate the projection exactly and in closed form. This yields explicit invariant-preserving integrators for a broad class of nonlinear systems, as well as pseudo-invariant-preserving schemes capable of preserving multiple invariants to arbitrarily high precision. The resulting methods are high-order and introduce negligible computational overhead relative to the base solver. When incorporated into adaptive solvers such as Dormand-Prince 8(5,3), they provide error-controlled, invariant-preserving, high-order time-stepping schemes. Numerical experiments on double-pendulum and Kepler ODEs as well as semidiscretised KdV and Camassa-Holm PDEs demonstrate substantial improvements in both accuracy and efficiency over standard approaches.