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

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 × 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 × 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.