Two-step fourth-order P-stable methods with phase-lag of order six for y″=(t,y)
Journal of Computational and Applied Mathematics
SIAM Journal on Numerical Analysis
Diagonally implicit Runge-Kutta-Nystro¨m methods for oscillatory problems
SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Iterative schemes for three-stage implicit Runge-Kutta methods
Applied Numerical Mathematics
Triangularly Implicit Iteration Methods for ODE-IVP Solvers
SIAM Journal on Scientific Computing
SDIRK methods for stiff ODEs with oscillating solutions
Journal of Computational and Applied Mathematics
Variable-order starting algorithms for implicit Runge-Kutta methods on stiff problems
Applied Numerical Mathematics
Journal of Computational and Applied Mathematics
Recent advances in linear analysis of convergence for splittings for solving ODE problems
Applied Numerical Mathematics
Accuracy and linear stability of RKN methods for solving second-order stiff problems
Applied Numerical Mathematics
A Code Based on the Two-Stage Runge-Kutta Gauss Formula for Second-Order Initial Value Problems
ACM Transactions on Mathematical Software (TOMS)
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We consider some important aspects about the implementation of high order implicit formulas (specially the Gauss methods) for solving second-order differential systems having high frequencies and small amplitudes superimposed. The choice of an appropriate iterative scheme is discussed in detail. Important topics about the predictors (initial guesses) are analyzed and a variable order strategy to select the best predictor at each integration step is supplied. A few numerical experiments on some standard test problems confirm the theory presented.