Novel high-order energy-preserving diagonally implicit Runge–Kutta schemes for nonlinear Hamiltonian ODEs

Abstract

We utilize the invariant energy quadratization approach to transform the nonlinear Hamiltonian ODE into an equivalent form which has a quadratic energy. Then the reformulation is discretized using a class of diagonally implicit Runge–Kutta schemes. The proposed schemes are proved to conserve a modified quadratic energy conservation law and converge with high-order accuracy. Numerical experiments for several ODEs are provided to illustrate the accuracy, convergence and conservative properties of the proposed method. Comparisons with the symplectic Runge–Kutta scheme and energy-preserving average vector field method are presented to demonstrate the energy-preserving property and accuracy of the proposed method.