# What are the problems that may arise by the application of Euler method?

What are the problems that may arise by the application of Euler method?

2 problems are described. One of them:

1.It is written that from from the convergence order it can be concluded that halving the error means halving the step-size and thus doubling the number of grid points. This leads to the doubling of computational costs which might cause an enormous effort.

Do we halve the step-size in Euler method? For my imagination for Euler method we just define a step size as we wish. As smaller we take as more precise will be our approximation. But where does "halving" come from? Where is my confusion?

Euler's method is used to numerically approximate the value of an ODE solution. Say we are given an ODE with no closed form solution ($y'+e^{-x^2}y=x^x$ for example), an initial value of $y(0)=1$ and we want to know $y(2)$. If we use a step size of $1$, we only need to do $2$ calculations to get our approximation but the error will be high. We can reduce the error by using a step size of $\frac{1}{2}$ but now we have to do $4$ calculations to get our approximation. We can repeat this indefinitely with each decrease in step size reducing the error but increasing the number of steps needed. Euler's method is a first order numerical method, meaning that the convergence order is proportional to the step size. Because of this, halving the step size will (roughly) halve the error.