Optimized High Order Explicit Runge-Kutta-Nyström Schemes

Marc Duruflé (1, 2) , Mamadou N'Diaye (3, 2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
3 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

In this paper, we are interested in solving a non-linear ordinary differential equation (ODE) of the type: y = f (t, y). For this ODE, high-order Runge-Kutta-Nyström have been proposed (see [1]). They are attractive because they are explicit, one-step methods and can be applied to a non-linear operator f. In [2], the stability condition (CFL) associated with these schemes have been studied for order 3, 4 and 5. In this paper, we extend this study for higher orders, and propose optimized coefficients with respect to the stability condition. With the obtained optimal CFL, these schemes are of practical interest for stiff problems where the stability condition is restrictive. These schemes are used for solving non-linear Maxwell's equations in 1-D: ε ∞ c 2 ∂ 2 E ∂t 2 + curl(curlE) + γ c 2 ∂ 2 ∂t 2 |E| 2 E = 0. The non-linearity is an instantaneous Kerr effect, where γ is the non-linear susceptibility. High-order finite elements are used in space to obtain the ODE to be solved. We are interested in the following ordinary differential equation (ODE)    y (t) = f (t, y(t)), ∀t > 0, y(0) = y 0 , y (0) = y 0. Marc Duruflé Magique-3D, INRIA Bordeaux Sud-Ouest,

Domains: Analysis of PDEs [math.AP]

Marc Duruflé, Mamadou N'Diaye. Optimized High Order Explicit Runge-Kutta-Nyström Schemes. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016 Selected Papers from the ICOSAHOM conference, June 27-July 1, 2016, Rio de Janeiro, Brazil , 2017. ⟨hal-01403338v1⟩