Highly efficient invariant-conserving explicit Runge-Kutta schemes for nonlinear Hamiltonian differential equations

Abstract

A unified framework of invariant-conserving explicit Runge-Kutta schemes for the nonlinear Hamiltonian ODEs and PDEs are proposed by utilizing the invariant energy quadratization technique. First, the nonlinear Hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant. For the nonlinear ODE, the reformulation is then discretized using a class of relaxation Runge-Kutta schemes. For the nonlinear PDE, the reformulation is first discretized in the space direction by adopting the Fourier pseudo-spectral discretization which preserves the semi-discrete quadratic conservation laws. Then the semi-discrete system is integrated in the time direction using the explicit relaxation Runge-Kutta method. The obtained method can conserve different quadratic conservation laws to machine accuracy, which improves the numerical stability during long time computations. Besides, the proposed method keeps the same convergence rate of the standard Runge-Kutta scheme when computed with time step size rescaling, and gives a rate of convergence reduced by at most one when computed without step size rescaling. Numerical experiments for several ODEs and PDEs are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis.