Modifications and Extensions
A simple modification of the Euler method which eliminates the stability problems noted in the previous section is the backward Euler method:
This differs from the (standard, or forward) Euler method in that the function is evaluated at the end point of the step, instead of the starting point. The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly.
Other modifications of the Euler method that help with stability yield the exponential Euler method or the semi-implicit Euler method.
More complicated methods can achieve a higher order (and more accuracy). One possibility is to use more function evaluations. This is illustrated by the midpoint method which is already mentioned in this article:
This leads to the family of Runge–Kutta methods.
The other possibility is to use more past values, as illustrated by the two-step Adams–Bashforth method:
This leads to the family of linear multistep methods.
Read more about this topic: Euler Method
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