Interpolants for Runge-Kutta formulas
ACM Transactions on Mathematical Software (TOMS)
Some practical Runge-Kutta formulas
Mathematics of Computation
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Order, stepsize and stiffness switching
Computing
Some Runge-Kutta formula pairs
SIAM Journal on Numerical Analysis
Numerical methods for ordinary differential systems: the initial value problem
Numerical methods for ordinary differential systems: the initial value problem
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Computer algebra (2nd ed.): systems and algorithms for algebraic computation
Computer algebra (2nd ed.): systems and algorithms for algebraic computation
Explicit Runge-Kutta pairs with one more derivative evaluation than the minimum
SIAM Journal on Scientific Computing
Symbolic derivation of Runge-Kutta methods
Journal of Symbolic Computation
Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods
ACM Transactions on Mathematical Software (TOMS)
Order stars and linear stability theory
Journal of Symbolic Computation
Modern computer algebra
Computer generation of numerical methods for ordinary differential equations
Recent trends in numerical analysis
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Second-order accurate projective integrators for multiscale problems
Journal of Computational and Applied Mathematics
| Hi-index | 0.98 |
Explicit Runge-Kutta schemes are the methods of choice for solving nonstiff systems of ordinary differential equations at low to medium tolerances. The construction of optimal formulae has been the subject of much research. In this article, it will be shown how to construct some low order formula pairs using tools from computer algebra. Our focus will be on methods that are equipped with local error detection (for adaptivity in the step size) and with the ability to detect stiffness. It will be demonstrated how criteria governing 'optimal' tuning of free parameters and matching of the embedded method can be accomplished by forming a constrained optimization problem. In contrast to standard numerical optimization processes our approach finds an exact (infinite precision) global minimum. Quantitative measures will be given comparing our new methods with some established formula pairs.