The structural method for Ordinary Differential Equations

TL;DR

This paper introduces a novel numerical method based on the structural approach for solving ODE systems, enabling simultaneous approximation of solutions and derivatives with high accuracy and spectral resolution.

Contribution

The paper develops a new structural method that computes solutions and derivatives of ODEs over multiple steps using linear relations, enhancing accuracy and spectral properties.

Findings

Provides highly accurate solutions with spectral resolution

Efficiently computes derivatives alongside solutions

Demonstrates improved accuracy over traditional methods

Abstract

We design and analyse a new numerical method to solve ODE system based on the structural method. We compute approximations of solutions together with its derivatives up to order $K$ by solving an entire block corresponding to $R$ time steps. We build the physical relations that connect the function and derivative approximations at each time step by using the ODE and its derivatives, and develop the structural equations that establish linear relations between the function and its derivative over the whole block of $R$ times steps. The non-linear system is solved and provide very accurate approximations with nice spectral resolution properties.