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Linear Multistep methods

A linear multistep method computes the next solution value from the values at several previous steps:

\(\alpha_k y_{n+k} + \alpha_{k-1} y_{n+k-1} + ... + \alpha_0 y_n = h ( \beta_k f_{n+k} + ... + \beta_0 f_n )\)

Note that different conventions for numbering the coefficients exist; the above form is used in NodePy. Methods are automatically normalized so that \(\\alpha_k=1\).

Instantiation

The follwing functions return linear multistep methods of some common types:

  • Adams-Bashforth methods: Adams_Bashforth(k)

  • Adams-Moulton methods: Adams_Moulton(k)

  • backward_difference_formula(k)

  • Optimal explicit SSP methods (elm_ssp2(k))

In each case, the argument \(k\) specifies the number of steps in the method. Note that it is possible to generate methods for arbitrary \(k\), but currently for large \(k\) there are large errors in the coefficients due to roundoff errors. This begins to be significant at 7 steps. However, members of these families with many steps do not have good properties.

More generally, a linear multistep method can be instantiated by specifying its coefficients \(\\alpha,\\beta\):

>> from nodepy import linear_multistep_method as lmm
>> my_lmm=lmm.LinearMultistepMethod(alpha,beta)

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