5. Numerical Simulation - Adam-Bashforth-Moulton-Method (A-B-M)

The requirements for the numerical integrator has to be accurate enough to keep the effects of numerical heating or cooling small and das to keep the computational effort small to achieve acceptable simulation times. The so called multistep method fulfil the requirements for the numerical integrator. This class of integrator methods make use of the information calculated in previous steps to update the integration scheme. The drawback is the loss in accuracy and the increased numbers of initial values needed to start in the integration scheme.

For the simulation we use the Adam-Bashforth-Moulton (A-B-M) predictor-corrector integration scheme of 6-th order. It is the set of first order differential equations of type $ \dot{\vec{y}}=\vec{f}(t,y). $ The A-B-M method combines the explicitness of the A-B method and the small discretization error of the A-M method.

\[ \vec{y}^{(P)}_{j+6} = \vec{y}_{j+5} + \frac{\Delta t}{1440} \left\{4277\vec{f}_{j+5} - 7923\vec{f}_{j+4} + 9982\vec{f}_{j+3} - 7298\vec{f}_{j+2} + 2877\vec{f}_{j+1} - 475\vec{f}_{j}\right\} \]
\[ \vec{y}_{j+6} = \vec{y}_{j+5} + \frac{\Delta t}{1440} \left\{ 475\vec{f}^{(P)}_{j+5} + 1427\vec{f}_{j+5} - 798\vec{f}_{j+4} + 482\vec{f}_{j+2} - 173\vec{f}_{j+2} + 27\vec{f}_{j+1} \right\} \]

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-- EvaMartenstein - 20 Apr 2015
Topic revision: r8 - 2016-05-10, AlexanderHenkel
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