Construction of variable-stepsize multistep formulas
Mathematics of Computation
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
A collocation formulation of multistep methods for variable step-size extensions
Applied Numerical Mathematics
Digital filters in adaptive time-stepping
ACM Transactions on Mathematical Software (TOMS)
Explicit, Time Reversible, Adaptive Step Size Control
SIAM Journal on Scientific Computing
Adaptive time-stepping and computational stability
Journal of Computational and Applied Mathematics - Special issue: International workshop on the technological aspects of mathematics
Adaptivity and computational complexity in the numerical solution of ODEs
Journal of Complexity
On quasi-consistent integration by Nordsieck methods
Journal of Computational and Applied Mathematics
Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation
Journal of Computational and Applied Mathematics
Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error Control
SIAM Journal on Scientific Computing
| Hi-index | 7.29 |
In linear multistep methods with variable step size, the method's coefficients are functions of the step size ratios. The coefficients therefore need to be recomputed on every step to retain the method's proper order of convergence. An alternative approach is to use step density control to make the method adaptive. If the step size sequence is smooth, the method can use constant coefficients without losing its order of convergence. The paper introduces this new adaptive technique and demonstrates its feasibility with a few test problems. The technique works in perfect agreement with theory for a given step density function. For practical use, however, the density must be generated with data computed from the numerical solution. We introduce a local error tracking controller, which automatically adapts the density to computed data, and demonstrate in computational experiments that the technique works well at least up to fourth-order methods.