Numerical solutions of ordinary differential equations using Spline-Integral Operator
Gustavo H. O. Salgado, João P. R. Romanelli
TL;DR
The paper addresses IVP truncation errors by introducing the Spline-Integral Operator (SIO), which blends a spline-based approximation $S_m(t_0,t,w)$ with the integral form $y(t)=y_0+ \int_{t_0}^t f(\tau,y(\tau))\,d\tau$. Using a fixed-point iteration to determine $w$ in $G_h(w)=y_0+\int_{t_0}^{t_0+h} f(t,S_m(t_0,t,w))\,dt$, the method is an implicit one-step scheme whose order is $m+1$ when the spline contains derivatives up to order $m-1$. Theoretical results establish contraction and convergence, yield an $\mathcal{O}(h^{m+1})$ accuracy for the $G_h$-approximation and $\mathcal{O}(h^{m+2})$ for the fixed point, and provide a stability region analysis via a linear test equation, showing that SIO(2) matches the implicit trapezoidal method and that higher $m$ improve stability. Numerical experiments with $m=3$ (SIO$(4)$) compare favorably to Taylor methods of the same order, using both analytic and Gauss-quadrature-based integration, and demonstrate robustness with respect to the integral evaluation technique. Overall, the SIO framework offers a high-order, implicit ODE solver that leverages spline-based approximations and fixed-point theory to achieve accurate, stable solutions with flexible computational options.
Abstract
In this work, we introduce a novel numerical method for solving initial value problems associated with a given differential. Our approach utilizes a spline approximation of the theoretical solution alongside the integral formulation of the analytical solution. Furthermore, we offer a rigorous proof of the method's order and provide a comprehensive stability analysis. Additionally, we showcase the effectiveness method through some examples, comparing with Taylor's methods of same order.