All explicit Runge-Kutta methods which are convergent with order q for explicit ODEs can be applied in this way to index-1 ODEs too. The order of the error in y t) and t) is q. [Pg.182]

This gives a good estimate provided At is small enough that the method is truly convergent with order p. This process can also be repeated in the same way Romberg s method was used for quadrature. [Pg.473]

It is well known that for sufficiently differentiable functions, the Euler method converges with order At (0(At)). This means that the total error after n integration steps can be made as small as wanted provided the interval At is small enough. Unfortunately this says nothing about the At to be used in order to [Pg.166]

If the multistep discretization defined by (J. 1.12) is consistent of order p and zero stable, then the method (5.2.6) converges with order p, i.e. the global error is [Pg.157]

In short, the error in the approximation obtained on [0, r] is reduced in direct proportion to the number of steps taken to cover this interval. Another way to say this is that e och, or, using the order notation, e = 0(h). Because the global error is of order h , where r= 1, we say that Euler s method is a first order method, or that it converges with order r = 1. [Pg.56]

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