An Essential Extension of the Finite-Energy Condition for ERKN Integrators Solving Nonlinear Wave Equations

This chapter is devotedFinite-energy condition to an essential extension of the finite-energy condition for extended Runge–Kutta–Nyström (ERKN) integrators when applied to nonlinear wave equations. We begin with an error analysis of ERKN integrators for multi-frequency highly oscillatory systems $$y''+My=f(y)$$ , where M is positive semi-definite, $$\Vert M\Vert \gg max\{1,\Vert \frac{\partial f}{\partial y}\Vert \}$$ . These highly oscillatory problems arise from the semi-discretisation of conservative or dissipative nonlinear wave equations. The structure of M and the initial conditions are dependent on the particular spatial discretisation. A finite-energy condition for the semi-discretisation of nonlinear wave equations is introduced and analysed. This is similar to the error analysis for Gaustchi-type methods of order two, where a finite-energy condition bounding amplitudes of high oscillations is satisfied by the solution. These ensure that the error bound for ERKN methods is independent of $$\Vert M\Vert $$ . Since stepsizes are not restricted by frequencies of M, large stepsizes can be employed by our ERKN integrators which may be of arbitrarily high order. The numerical experiments presented in this chapter demonstrate that our results are really promising, and consistent with our analysis and predictions.