Improved uniform error bounds on parareal exponential algorithm for highly oscillatory systems

For the well known parareal algorithm, we formulate and analyse a novel class of parareal exponential schemes with improved uniform accuracy for highly oscillatory system \(\ddot{q}+\frac{1}{\varepsilon ^2}M q =\frac{1}{\varepsilon ^{\mu }}f(q)\) with \(\mu =0\) or 1. The solution of this considered system propagates waves with wavelength at \(\mathcal {O} (\varepsilon )\) in time and the value of \(\mu \) corresponds to the strength of nonlinearity. This brings significantly numerical burden in scientific computation for highly oscillatory systems with \(0<\varepsilon \ll 1\). The new proposed algorithm is formulated by using some reformulation approaches to the problem, Fourier pseudo-spectral methods, and parareal exponential integrators. The fast Fourier transform is incorporated in the implementation. We rigorously study the convergence, showing that for nonlinear systems, the algorithm has improved uniform accuracy \(\mathcal {O}\big ( \varepsilon ^{(2k+3)(1-\mu )}\Delta t^{2k+2}+\varepsilon ^{5(1-\mu )}\delta t^4\big )\) in the position and \(\mathcal {O}\big ( \varepsilon ^{(2k+3)(1-\mu )-1}\Delta t^{2k+2}+\varepsilon ^{4-5\mu }\delta t^4\big )\) in the momenta, where k is the number of parareal iterations, and \(\Delta t\) and \(\delta t\) are two time stepsizes used in the algorithm. The energy conservation is also discussed and the algorithm is shown to have an improved energy conservation. Numerical experiments are provided and the numerical results demonstrate the improved uniform accuracy and improved energy conservation of the obtained integrator through four Hamiltonian differential equations including nonlinear wave equations.