# Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations

## TL;DR

This paper introduces an efficient implementation of symplectic implicit Runge-Kutta methods using a novel linear system rewriting technique, reducing computational effort and round-off errors in numerical solutions of differential equations.

## Contribution

It proposes a new way to solve the linear systems in symplectic IRK methods by rewriting them as smaller systems, improving efficiency and numerical stability.

## Key findings

- Reduced round-off error propagation demonstrated
- Efficient LU decomposition approach for linear systems
- Implementation shows improved numerical stability in experiments

## Abstract

We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular attention to symmetric symplectic IRK schemes (such as collocation methods with Gaussian nodes). For a $s$-stage IRK scheme used to integrate a $d$-dimensional system of ordinary differential equations, the application of simplified versions of Newton iterations requires solving at each step several linear systems (one per iteration) with the same $sd \times sd$ real coefficient matrix. We propose rewriting such $sd$-dimensional linear systems as an equivalent $(s+1)d$-dimensional systems that can be solved by performing the LU decompositions of $[s/2] +1$ real matrices of size $d \times d$. We present a C implementation (based on Newton-like iterations) of Runge-Kutta collocation methods with Gaussian nodes that make use of such a rewriting of the linear system and that takes special care in reducing the effect of round-off errors. We report some numerical experiments that demonstrate the reduced round-off error propagation of our implementation.