Numerical solutions of differential equations

• Note that this is not tested in MAT22B

Numerical solutions of IVPs involve using taylor series and the definition of derivatives:

${\frac {dy}{dx}}(x=x_{0})=\lim _{h\rightarrow 0}{\frac {y(x_{0}+h)-y(x_{0})}{h}}$

Since we know one initial value of the function, we can use the above property to iteratively approximate the surrounding values with a small step (h).

Classification

Numerical solutions are evaluated by three criterias:

1. Convergence checks if the solution is correct as h approaches 0
2. Accuracy measures how good the approximation is
3. Stability I am not very familiar with.

Accuracy is measured by order of h in the error. A higher order usually means that with each reduction in h, there is a greater increase in accuracy.