What is application of Euler method?

Question

What are the application fields of the Euler forward and backward methods?

What does the big O notation mean, in the case of Euler method $O(h^2)$ and $O(h)$?

Answer 1

These are methods for the numerical solution of ordinary differential equations, assuming that they were transformed to a system of order 1.

The Euler methods are the most simple explicit and implicit Runge-Kutta methods. They should only be used as educational examples.

Explicit Runge-Kutta methods are used for ODE that are mostly linear, with little variation in the coefficient magnitude (or more precisely, a low condition number). At the other end you have ODE that are highly non-linear or that have solution components of drastically different speed. These require implicit methods.

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For a differential equation $y'=f(y)$, $y(0)=y_0$ with exact solution $y(t)$ the big-O-notation says that the error of the Euler method at the first iteration step, $y_1=y_0+h\,f(y_0)$, as representative example of every other iteration step or initial point, is quadratic in $h$, that is, $y_1-y(t_0+h)=O(h^2)$, where $y(t_0)=y_0$ and $(t_0,y_0)$ is arbitrary in the domain of the ODE.

Note that the time $t_1=t_0+h$ of the first value varies with the step length $h$.

For a fixed time $T$ and $n=[T/h]$ varying with $h$, the accumulated error is $y_n-y(nh)=n\cdot O(h^2)=O(h)$. As $y(T)-y(nh)=O(h)$, this error order remains valid if the last numerical value is compared with the fixed exact value $y(T)$.

The more precise formula for this error is $O((e^{LT}-1)h)$ with $L$ the Lipschitz constant of $f$.

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See wikipedia on the (forward) Euler method, backward Euler method and the Landau notation.

Please use these articles to refine your question to more detailed problems.