MTH3007b Lecture 2

Me, in the lecture

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The beginning of this lecture started with a lot of recap of the last lecture and basic programming; like using integers for countable numbers and floats for non-countable numbers. Then, we quickly covered round-off errors - exactly what you’d assume, errors that occur due to rounding.

After finishing that off, we dived into actual numerical methods: higher order integration methods…

Runge-Kutta Methods

For each Euler method, their global error is $O(\Delta t)$ and order is one. We can generalise these however, to get the Runge-Kutta methods:

$$y_{i+1}\approx y_i+\Delta t\cdot \varphi (t_i,y_i,\Delta t)$$

Where $\varphi$ is called the increment function, and the simplest case of this is $\varphi (t_i,y_i,\Delta t)=g(t_i,y_i)$ - the forward Euler method.

The second order Runge-Kutta method, also called the Midpoint method, which evaluates a function $g$ at each midpoint between timesteps:

$$y_{i+1}\approx y_i+\Delta t\cdot g\left(t_i+\frac{\Delta t}{2},y_i+g(t_i,y_i)\frac{\Delta t}{2}\right)$$

Alternatively, we can use a modified version of this evaluating two-thirds through the interval instead of halfway. This is called the Ralston method:

$$y_{i+1}\approx y_i+\Delta t\cdot \left(\frac{1}{4}g(t_i,y_i)+\frac{3}{4}g\left(t_i+\frac{2}{3}\Delta t,y_i+\frac{2}{3}g(t_i,y_i)\Delta t\right)\right)$$

However, this is a bit easier to implement by breaking it into intermediate variables, namely $k_1=g(t_i,y_i)$ and $k_2=g\left(t_i+\frac{2}{3}\Delta t,y_i+\frac{2}{3}\Delta t\cdot k_1\right)$:

$$y_{i+1}\approx y_i+\Delta t\cdot \left(\frac{1}{4}{k}_1+\frac{3}{4}{k}_2\right)$$

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Pre-Lecture Notes from University Notes

• Recalling Euler method and applications.
• Discussing function value types: integers, floats, etc.
• Exploring round-off errors.
• Exploring higher order integration methods, like Runge-Kutta methods, midpoint method, and Ralston’s method.