Linear Stability of Partitioned Runge-Kutta Methods
SIAM Journal on Numerical Analysis
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We propose a “Newton-Taylor” iteration for solving the implicit equations of symplectic Runge-Kutta methods, using the Jacobian of the vector field and matrix-vector multiplications whose extra cost for certain structured problems is negligible. The structure of Hamiltonian ODEs allows this very simple iteration to be effective. The iteration reduces the number of vector field evaluations almost to that of Newton's method, often only one or two per time step, making symplectic Runge-Kutta methods more efficient even at relatively large time steps.