Fixed versus variable order Runge-Kutta
ACM Transactions on Mathematical Software (TOMS) - The MIT Press scientific computation series
Two FORTRAN packages for assessing initial value methods
ACM Transactions on Mathematical Software (TOMS)
Algorithm 719: Multiprecision translation and execution of FORTRAN programs
ACM Transactions on Mathematical Software (TOMS)
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ACM Transactions on Mathematical Software (TOMS)
Chebyshev collocation methods for fast orbit determination
Applied Mathematics and Computation
On Taylor Series and Stiff Equations
ACM Transactions on Mathematical Software (TOMS)
Solving Ordinary Differential Equations Using Taylor Series
ACM Transactions on Mathematical Software (TOMS)
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Communications of the ACM
Continuous numerical methods for ODEs with defect control
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
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SIAM Journal on Scientific Computing
High-Order Stiff ODE Solvers via Automatic Differentiation and Rational Prediction
WNAA '96 Proceedings of the First International Workshop on Numerical Analysis and Its Applications
A one-step 7-stage Hermite-Birkhoff-Taylor ODE solver of order 11
Journal of Computational and Applied Mathematics
Reducing rounding errors and achieving Brouwer's law with Taylor Series Method
Applied Numerical Mathematics
Algorithm 924: TIDES, a Taylor Series Integrator for Differential EquationS
ACM Transactions on Mathematical Software (TOMS)
Exponential Taylor methods: Analysis and implementation
Computers & Mathematics with Applications
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This paper presents an analysis of the Taylor method for the numerical solution of ODEs when a very high precision of the solution is required. Some theoretical properties of the Taylor method are considered. From the practical point of view a variable-stepsize variable-order (VSVO) scheme is presented and its utility is discussed with several examples. To reach the goal of high precision the use of multiprecision libraries is considered. Finally, some numerical tests based on the test problems given in [1] and on a set of important problems in dynamical systems and astrodynamics are presented showing the benefits of the VSVO formulation, especially for high-precision demands, compared with a well established Runge-Kutta code.