## VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (5 & 8 March) in your usual lab time.

Tuesday 10:05-11:45 will run 10:05 to 11:55

Tuesday 14:20-16:00 will run 14:20-16:10

Friday 09:05-10:45 will run 09:05-10:55

In the Week 5 VBA we worked on Lab 3. Those of us who did not finish the lab are advised to finish it outside class time, and are free to email me on their work if they are unsure if they are correct or not.

## Written Assessment 1

Written Assessment 1 takes place Tuesday 12 March at 09:00 in the usual lecture venue.

Here is a copy of last year’s assessment. This should give you an idea of the length and format but not what questions are coming up – and replaces Section 1.6.1 of the manual.

However there are far more things I could examine.

Roughly, everything up to but not including Runge Kutta Methods (p.64). Some examples of questions I could ask include:

### ODEs in Engineering

p.13, Examples 1-4; p.15, Q.1-4

### General Theory

Example, p. 15; p.34 Example

### Maclaurin/Taylor Series

Examples 1 & 2 on p. 24; Q. 1 on p.27

### Euler Method

p.29, Examples 1-4; p. 38, Q.1-5, 8-9

### Three Term Taylor Method

p. 35, Examples 1-2; p.39, Q.7, 10-14

### Heun’s Method

p.38, Q. 6; p. 42, Examples 1-2l p. 47, Q.4-5

### Second Order Differential Equations

p.50, Example. p.51, Example. p.55, Q. 1-3, 5-14

## Runge Kutta Methods Summary

Given an initial value problem (IVP): $\displaystyle \frac{dy}{dx}=F(x,y)\,,\,\,\qquad y(x_0)=y_0$,

I want you to understand the following:

First of all:

• The Two Term Taylor Method is the Euler Method and uses lines to approximate $y(x_n)$.
• If the solution of the IVP is a line, the error in the Euler Method aka the Two Term Taylor Method is zero. Note that the the second derivative of a line is zero.
• The global error for Euler’s Method $\mathcal{O}(h)$ and so Euler’s Method aka the Two Term Taylor Method is said to a first order method (because the error is $h^1$).
• The Three Term Taylor Method uses parabolas, degree two polynomials, to approximate $y(x_n)$.
• If the solution of the IVP is a parabola, the error in the Three Term Taylor Method is zero. Note that the third derivative of a parabola is zero.
• The global error for the Three Term Taylor Method is $\mathcal{O}(h^2)$ and so the Three Term Taylor Method is said to be a second order method.
• If $F(x,y)$ is dependent on $y$, then the Three Term Taylor Method requires implicit differentiation.
• With further implicit differentiation, we have the $(n+1)$-Term Taylor Method which uses a degree $n$ polynomials to approximate $y(x_n)$.
• If the solution to the IVP is an a degree $n$ polynomial, the error in the $(n+1)$-Term Taylor Method is zero. Note that the $(n+1)$-st derivative of a degree $n$ polynomial is zero.
• The global error for the $(n+1)$-Term Taylor Method is $\mathcal{O}(h^n)$ and so the $(n+1)$-Term Taylor Method is said to be a $n$-th order method.

The Euler Method is the first order Runge-Kutta Method.

Now what we want for a second order Runge-Kutta Method is that:

• Like the Three Term Taylor Method, it has zero error when the solution is a parabola.
• Like the Three Term Taylor Method, the global error is $\mathcal{O}(h^2)$.

It does this by taking a weighted average of two slopes: $k_1=F(x_i,y_i)$ the slope at $(x_i,y_i)$, and $k_2=F(x_i+ph,y_i+phk_1)$, the slope at any other point between $x_i$ and $x_i+h$.

Two such schemes are presented in the notes.

Now what we want for an $n$-th order Runge-Kutta is that:

• Like the $(n+1)$-Term Taylor Method, is has zero error when the solution is a degree $n$ polynomial.
• Like the $(n+1)$-Term Taylor Method, the global error is $\mathcal{O}(h^n)$.

It does this by taking a weighted average of $n$ slopes: $k_1,\dots, k_n$, where $k_1=F(x_i,y_i)$ the slope at $(x_i,y_i)$, and the other $k_j$ are slopes at $n-1$ points between $x_i$ and $x_i+h$.

See p.108-109 for other Runge-Kutta Theory exercises.

## Week 5

In the morning class we finished looking at second order differential equations.

In the afternoon, we will looked of Runge-Kutta Methods. A summary is given above and you will want to understand all of this for Written Assessment 2.

In VBA we had MCQ IV and looked at Lab 4, on Second Order Differential Equations.

## Week 6

We will look at boundary value problems (in particular the Shooting Method and Goal Seek).

In VBA we have MCQ V and VBA Assessment 1.

## Assessment

The following is a proposed assessment schedule:

1. Week 6, 20% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 20 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 20% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 40% Written Assessment(s), Based on Weeks 1-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Ungraded Concept MCQ League Table

To add a bit of interest to the Ungraded Concept MCQs, I will keep a league table.

Unless you are excelling, you are identified by the last five digits of your student number. AW is the number of attendance warnings received. 