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The relation between local and global truncation errors is slightly different from in the simpler setting of one-step methods. For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors.
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In general, a method with (+) LTE (local truncation error) is said to be of kth order. The region of absolute stability for the backward Euler method is the complement in the complex plane of the disk with radius 1 centered at 1, depicted in the figure. [ 4 ]
For example, the shooting method (and its variants) or global methods like finite differences, [3] Galerkin methods, [4] or collocation methods are appropriate for that class of problems. The Picard–Lindelöf theorem states that there is a unique solution, provided f is Lipschitz-continuous .
The ODE solver in many programming languages uses this procedure as the default strategy for adaptive stepsize control, which adds other engineering parameters to make the system more stable. See also
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Dormand–Prince is the default method in the ode45 solver for MATLAB [4] and GNU Octave [5] and is the default choice for the Simulink's model explorer solver. It is an option in Python's SciPy ODE integration library [6] and in Julia's ODE solvers library. [7] Implementations for the languages Fortran, [8] Java, [9] and C++ [10] are also ...
Verlet integration (French pronunciation:) is a numerical method used to integrate Newton's equations of motion. [1] It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics.