Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Consistency and stability for some nonnegativity-conserving methods
Applied Numerical Mathematics
Introduction to Maple (2nd ed.)
Introduction to Maple (2nd ed.)
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The numerical treatment of the ODE initial value problems is an intensively researched field. Recently the qualitative algorithms, such as monotonicity and positivity preserving algorithms are in the focus of investigation. For the dynamical systems the energy conservative algorithms are very important. In the case of Hamiltonian system the symptectic algorithms are very effective. This kind of algorithm is not adaptive, but doubtless they are powerful. The high-efficiency computers and the computer algebraic software systems allow us to create efficient adaptive energy conservative numerical algorithm for solving ODE initial value problems. In this article an adaptive numerical-analytical algorithm is suggested which very effectively can be applied for Hamiltonian systems, but the idea of construction can be adaptable for other initial value problems too, where there are some quantity preserved in time. The idea and the efficiency of the proposed algorithm will be presented by simple examples, such as the Lotka-Volterra and linear oscillator problems.