Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations

Abstract: This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including quasi-Fermat's theorem and the 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 first-order consistency and second-order local convergence.