glutynowy44

2022-07-07

I don't quite understand the discretisation error of Euler's method
$L\left(t,h\right)=\frac{x\left(t+h\right)-x\left(t\right)}{h}-f\left(t,x\left(t\right)\right)$
What I don't understand is: $\frac{x\left(t+h\right)-x\left(t\right)}{h}$ is the "gradient" and $f\left(t,x\left(t\right)$ is the value of the function. How can we get the error by subtracting the former by the latter?

Caiden Barrett

Expert

You assume that $x\left(t\right)$ is an exact solution to $\stackrel{˙}{x}=f\left(t,x\right)$ and then apply Taylor formulas.