Refining Euler and fourth-order Runge-Kutta methods using curvature-based adaptivity
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Abstract
This paper introduces curvature-adaptive variants of both the Euler and Runge-Kutta methods for solving ordinary differential equations (ODEs). By incorporating local curvature information into step size selection, the proposed Curvature-Adaptive Euler (CAE) and Curvature-Adaptive fourth-order Runge-Kutta (CARK4) methods achieve a superior balance between accuracy and computational efficiency compared to their fixed-step counterparts. The unified curvature adaptation framework presented in this work enhances classical ODE solvers by leveraging the geometric properties of the solution curve, resulting in more efficient numerical integration schemes.
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