General linear methods for ordinary differential equations – p. 1/46 Solving ordinary differential equations numerically is, even today, still a great challenge. This applies especially to stiff differential equations and to closely related problems involving algebraic constraints (DAEs). Although the problem seems to be solved — there are already highly… (More)
We describe the construction of explicit general linear methods of order p and stage order q = p with s = p + 1 stages which achieve good balance between accuracy and stability properties. The conditions are imposed on the coefficients of these methods which ensure that the resulting stability matrix has only one nonzero eigenvalue. This eigenvalue depends… (More)
Implicit Runge-Kutta methods are considered which combine the single-implicitness or diagonal-implicitness property with a zero first row in the coefficient matrix. Acceptable stability for stiff problems is retained by requiring the last stage of a step to be identical to the output value. This requirement, which corresponds to the FSAL property for… (More)
This version is made available in accordance with publisher policies. Please cite only the published version using the reference above. Abstract. G-symplectic general linear methods are designed to approximately preserve sym-plectic invariants for Hamiltonian systems. In this paper, the properties of G-symplectic methods are explored computationally and… (More)
The concept of effective order allows for the possibility that the result computed in a Runge–Kutta step is an approximation to some quantity more general than the actual solution at a step point. This generalization is applied here to singly-implicit methods. The limitation that requires severe and inconvenient restrictions on the abscissae in the method… (More)
A new representation for diagonally implicit multistage integration methods (DIMSIMs) is derived in which the vector of external stages directly approximates the Nordsieck vector. The methods in this formulation are zero-stable for any choice of variable mesh. They are also easy to implement since changing step-size corresponds to a simple rescaling of the… (More)
We describe the construction of diagonally implicit multistage integration methods of order and stage order p = q = 7 and p = q = 8 for ordinary diierential equations. These methods were obtained using state-of-the-art optimization methods, particularly variable-model trust-region least-squares algorithms.
Certain lmphclt Runge-Kutta methods are capable of being transformed into a form which makes the modified Newton iterates in their implementation capable of efficient computation. For the class of such methods considered, the transformations are given exphcltly, and ~t is shown how error estimates, as well as initial iterates for a succeeding step, can be… (More)