2013/01/20 by Richard A. Norton, Norton, Richard A., G. Quispel +1
Computer Science · Mathematics · #37M99 #65D30 #65L20 #70B10 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1301.4717
openalex publication_date 2013/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we consider discrete gradient methods for approximating the\nsolution and preserving a first integral (also called a constant of motion) of\nautonomous ordinary differential equations. We prove under mild conditions for\na large class of discrete gradient methods that the numerical solution exists\nand is locally unique, and that for arbitrary p \∈ \ℕ we may\nconstruct a method that is of order p. In the proofs of these results we also\nshow that the constants in the time step constraint and the error bounds may be\nchosen independently from the distance to critical points of the first\nintegral. In the case when the first integral is quadratic, for arbitrary p\n\∈ \ℕ, we have devised a new method that is linearly implicit at each\ntime step and of order p. This new method has significant advantages in terms\nof efficiency. We illustrate our theory with a numerical example.\n