Explicit Euler Method

Using the Finite Difference Method, we can solve Ordinary Differential Equations (ODEs). For example with the Euler method, where we first replace the derivatives with finite differences.

For an initial value problem, the ODE can be written as:

$$\frac{f(x+\Delta x)-f(x)}{\Delta x}\approx g(x,f(x))⟹\boxed{f(x+\Delta x{)}_{\text{approx}}\approx f(x)+\Delta x\cdot g(x,f(x))}$$

This is called the Explicit Euler method, or the forward Euler method, and we can calculate the total number of integration steps as $N_{\text{int}}=\frac{|x_{\text{end}}-x_{\text{start}}|}{\Delta x}$ and hence $x_{\text{end}}=x{N}_{\text{int}}=x_0+N_{\text{int}}\Delta x$. A smaller $\Delta x$ we hence naturally improve the accuracy, but at a computational cost.