Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations
This work provides a new theoretical tool (specular differentiation) for numerical ODE solvers, but the practical impact is limited to specific cases and the novelty is incremental.
The paper introduces specular differentiation and uses it to develop numerical schemes for first-order ODEs, achieving second-order consistency and convergence for one scheme and zero local truncation error for elliptic trajectories.
This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses.