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.

翻译：本文提出了一种用于数值求解微分方程并保持守恒量的通用框架。与标准投影方法类似，我们将任意基础积分器投影至守恒流形上，但本方法利用齐次对称性，以闭合形式精确计算投影过程。该方法为广泛非线性系统提供了显式守恒积分器，并能构造伪守恒格式，以任意高精度保持多个守恒量。所得方法具有高阶精度，且相对于基础求解器仅引入可忽略的计算开销。当集成至自适应求解器（如Dormand-Prince 8(5,3)）时，可形成误差可控、守恒保持的高阶时间步进格式。在双摆与开普勒常微分方程以及半离散化KdV和Camassa-Holm偏微分方程上的数值实验表明，该方法在精度与效率上均较传统方法有显著提升。