Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
A general class of two-step Runge-Kutta methods for ordinary differential equations
SIAM Journal on Numerical Analysis
Digital filters in adaptive time-stepping
ACM Transactions on Mathematical Software (TOMS)
Nordsieck representation of two-step Runge--Kutta methods for ordinary differential equations
Applied Numerical Mathematics
Two-step modified collocation methods with structured coefficient matrices
Applied Numerical Mathematics
Computers & Mathematics with Applications
Two-step diagonally-implicit collocation based methods for Volterra Integral Equations
Applied Numerical Mathematics
Two-step modified collocation methods with structured coefficient matrices
Applied Numerical Mathematics
Numerical search for algebraically stable two-step almost collocation methods
Journal of Computational and Applied Mathematics
Exponentially fitted singly diagonally implicit Runge-Kutta methods
Journal of Computational and Applied Mathematics
Search for efficient general linear methods for ordinary differential equations
Journal of Computational and Applied Mathematics
P-stable general Nyström methods for y˝=f(y(t))
Journal of Computational and Applied Mathematics
Order conditions for General Linear Nyström methods
Numerical Algorithms
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Abstract: We describe a class of two-step continuous methods for the numerical integration of initial-value problems based on stiff ordinary differential equations (ODEs). These methods generalize the class of two-step Runge-Kutta methods. We restrict our attention to methods of order p=m, where m is the number of internal stages, and stage order q=p to avoid order reduction phenomenon for stiff equations, and determine some of the parameters to reduce the contribution of high order terms in the local discretization error. Moreover, we enforce the methods to be A-stable and L-stable. The results of some fixed and variable stepsize numerical experiments which indicate the effectiveness of two-step continuous methods and reliability of local error estimation will also be presented.