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Ruoqiao Li

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Proceedings Article | 22 April 2022 Paper

Equivalence between discrete Galerkin method and Runge-Kutta methods for initial value ordinary differential equations

Ruoqiao Li

Proceedings Volume 12163, 121633V (2022) https://doi.org/10.1117/12.2628102

KEYWORDS: Differential equations, Ordinary differential equations, Numerical analysis, Mathematical modeling, Finite difference methods, Error analysis, Calculus

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It is indispensable to seek approximations to ordinary differential equations that are more accurate than the Euler method to increase the speed of convergence of Euler's method. Euler's method is intuitively a linear Taylor polynomial approximation. It is reasonable to design higher-order Taylor approximations; as a result, a family of high-order methods emerges. It is required that higher regularity of the solution derive a Taylor method. Higher-order expressions are obtained by differentiating the differential equation from the solution itself. It is usually time-consuming to get higher-order derivative expressions. The idea behind Runge–Kutta's techniques is to approximate the derivative terms by combining compositions of the function of the differential equation. In this paper, we review the Taylor method and Runge-Kutta methods. We give a detailed proof of how the discrete Galerkin method is equivalent to some implicit Runge-Kutta method. These methods are stable and easy to implement.

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