Abstract:

This paper proposes a new structure-preserving numerical algorithm for autonomous ordinary differential equation systems with significant applications in engineering and physics. Based on the framework of general linear methods, the algorithm modifies coefficients of conventional methods through relaxation parameters, achieving precise preservation of key geometric features including Hamiltonian structure and energy conservation laws. The constructed relaxed general linear method combines advantages of linear-implicit formulation and high-order accuracy. Theoretical analysis and numerical experiments on typical physical systems-such as the Landau-Lifshitz equation, Kepler equation, and sine-Gordon equation-demonstrate that the relaxed algorithm significantly improves computational efficiency while maintaining the absolute stability region of original methods, and exhibits superior performance in structure preservation and energy conservation. This method extends the relaxation technique to the framework of general linear methods, constructing a class of linearly implicit structure-preserving algorithms suitable for autonomous ordinary differential equation systems, which can be employed for long-time, high-precision simulations of relevant physical models.

Key words: ordinary differential equations, general linear method, relaxation technique, structure-preserving algorithm