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I am not sure has the following observation been made:

When the Jacobi Method is used to approximate the solution of Laplace’s Equation, if the initial temperature distribution is given by $T^0(\mathbf{x})$, then the iterations $T^{\ell}(\mathbf{x})$ are also approximations to the solution, $T(\mathbf{x},t)$, of the Heat Equation, assuming the initial temperature distribution is $T^0(\mathbf{x})$.

I first considered such a thought while looking at approximations to the solution of Laplace’s Equation on a thin plate. The way I implemented the approximations was I wrote the iterations onto an Excel worksheet, and also included conditional formatting to represent the areas of hotter and colder, and the following kind of output was produced:

Let me say before I go on that this was an implementation of the Gauss-Seidel Method rather than the Jacobi Method, and furthermore the stopping rule used was the rather crude $|T^{\ell+1}_{i,j}-T^{\ell}_{i,j}|<\varepsilon$.

However, do not the iterations resemble the flow of heat from the heat source on the bottom through the plate? The aim of this post is to investigate this further. All boundaries will be assumed uninsulated to ease analysis.

## Discretisation

Consider a thin rod of length $L$. If we mesh the rod into $n$ pieces of equal length $\Delta x=L/n$, we have discretised the rod, into segments of length $\Delta x$, together with ‘nodes’ $0=x_0<\Delta x=x_1<2\Delta x=x_2<\cdots.

Suppose are interested in the temperature of the rod at a point $x\in[0,L]$, $T(x)$. We can instead consider a sampling of $T$, at the points $x_i$:

$\displaystyle T(x_i)=T(i\Delta x)=:T_i$.

Similarly we can mesh a plate of dimensions $W\times H$ into an $n\times m$ rectangular grid, with each rectangle of area $\Delta x\Delta y$, where $n\Delta x=W$ and $m\Delta y=H$, together with nodes $x_{i,j}=(i\Delta x,j\Delta y)$, and we can study the temperature of the plate at a point $\mathbf{x}\in[0,W]\times [0,H]$ by sampling at the points $x_{i,j}$:

$\displaystyle T(x_{i,j})=T(i\Delta x,j\Delta y)=:T_{i,j}$.

We can also mesh a box of dimension $W\times D\times H$ into an $n_1\times n_2\times n_2$ 3D grid, with each rectangular box of volume $\Delta x\Delta y\Delta z$, where $n_1\Delta x=W$, $n_2\Delta y=D$, and $n_3\Delta z=H$, together with nodes $x_{i,j,k}=(i\Delta x,j\Delta y,k\Delta z)$, and we can study the temperature of the box at the point $\mathbf{x}\in [0,W]\times [0,D]\times [0,H]$ by sampling at the points $x_{i,j,k}$:

$\displaystyle T(x_{i,j,k})=T(i\Delta x,j\Delta y,k\Delta z)=:T_{i,j,k}$.

## Finite Differences

How the temperature evolves is given by partial differential equations, expressing relationships between $T$ and its rates of change.

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Week 13

### Catch Up/Revision of Lab 8 Material

The final assessment, based on Lab 8, takes place 11:00, Tuesday 12 May.

If you have not yet done so, you will undertake the learning described in Week 10. Perhaps you should also look at the theory exercises described in Week 10.

The final assessment will ask you to do the following:

Consider:

$\displaystyle k\cdot \frac{\partial^2 T}{\partial x^2}=\frac{\partial T}{\partial t}$      (1)

For one (i.e. $x_1$ or $x_2$ or $x_3$) or all (i.e. general $x_i$), use equations (2) and (3) to write (1) in the form:

$T^{\ell+1}_i=f(T_j^\ell),$

i.e. find the temperature at node $i$ at time $\ell+1\equiv(\ell+1)\cdot \Delta t$ in terms of the temperatures at the previous time $\ell\equiv \ell\cdot \Delta t$. You may take $\Delta t=0.5$.

\begin{aligned} \left.\frac{dy}{dx}\right|_{x_i}&\approx \frac{y(x_{i+1})-y(x_i)}{\Delta x}\qquad(2) \\ \left.\frac{d^2y}{dx^2}\right|_{x_i}&\approx \frac{y(x_{i+1})-2y(x_i)+y(x_{i-1})}{(\Delta x)^2}\qquad(3) \end{aligned}

so if you want to send on this work for feedback please do so.

If you submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Submit work — VBA or Theory, catch-up or revision — based on Week 10 to the Lab 8 VBA/Theory Catch-up/Revision II assignment on Canvas by midnight 9 May. If you have any further questions, I am answering emails seven days a week. Email before midnight Monday 11 May to be guaranteed a response Tuesday 12 May. I cannot guarantee that I answer emails sent on Tuesday (although of course I will try).

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Week 12/13

### 11:00 Tuesday 28 April, Week 12: Assessment Based on Lab 7

Students can submit work — VBA or Theory, catch-up or revision — based on Week 9 Lab 7 VBA/Theory Catch-up/Revision II assignment by today, Saturday 25 April.

If you have any further questions, I am answering emails seven days a week. Email before midnight Monday 27 April to be guaranteed a response Tuesday 28 April. I cannot guarantee that I answer emails sent on Tuesday morning (although of course I will try).

### Catch Up/Revision of Lab 8 Material

If you have not yet done so, you will undertake the learning described in Week 10.

If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 10.

If you have already conducted this learning, and submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Students will be able submit work — VBA or Theory, catch-up or revision — based on Week 10 to a Lab 8 VBA/Theory Catch-up/Revision assignment on Canvas (due dates 3 May and 9 May).

## Week 14

### 11:00 Tuesday 12 May, Week 14: Assessment Based on Lab 8

This assessment will ask you to do the following:

Consider:

$\displaystyle k\cdot \frac{\partial^2 T}{\partial x^2}=\frac{\partial T}{\partial t}$      (1)

For one (i.e. $x_1$ or $x_2$ or $x_3$) or all (i.e. general $x_i$), use equations (2) and (3) to write (1) in the form:

$T^{\ell+1}_i=f(T_j^\ell),$

i.e. find the temperature at node $i$ at time $\ell+1\equiv(\ell+1)\cdot \Delta t$ in terms of the temperatures at the previous time $\ell\equiv \ell\cdot \Delta t$. You may take $\Delta t=0.5$.

\begin{aligned} \left.\frac{dy}{dx}\right|_{x_i}&\approx \frac{y(x_{i+1})-y(x_i)}{\Delta x}\qquad(2) \\ \left.\frac{d^2y}{dx^2}\right|_{x_i}&\approx \frac{y(x_{i+1})-2y(x_i)+y(x_{i-1})}{(\Delta x)^2}\qquad(3) \end{aligned}

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Week 11

### 20% VBA Assessment Based on Lab 6

Thank you to everyone for completing the assignment.

### Catch Up/Revision

You are advised to catch-up on the learning described in Week 9.

If you have already conducted this learning, and submitted a Lab 7 either back before 30 March, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback. If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 9.

Students can submit work — VBA or Theory, catch-up or revision — based on Week 9 to the Lab 7 VBA/Theory Catch-up/Revision I assignment on Canvas (by Sunday 19 April), or Lab 7 VBA/Theory Catch-up/Revision II assignment by Saturday 25 April.

## Week 12/13

### Catch Up/Revision

If you have not yet done so, you will undertake the learning described in Week 10.

If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 10.

If you have already conducted this learning, and submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Students will be able submit work — VBA or Theory, catch-up or revision — based on Week 10 to a Lab 8 VBA/Theory Catch-up/Revision assignment on Canvas.

I may have two due dates. Perhaps Sunday 3 May and Saturday 9 May.

## PROVISIONAL: Tuesday 12 May, Week 14: Assessment Based on Lab 8

This assessment will ask you to do the following:

Consider:

$\displaystyle k\cdot \frac{\partial^2 T}{\partial x^2}=\frac{\partial T}{\partial t}$      (1)

For one (i.e. $x_1$ or $x_2$ or $x_3$) or all (i.e. general $x_i$), use equations (2) and (3) to write (1) in the form:

$T^{\ell+1}_i=f(T_j^\ell),$

i.e. find the temperature at node $i$ at time $\ell+1\equiv(\ell+1)\cdot \Delta t$ in terms of the temperatures at the previous time $\ell\equiv \ell\cdot \Delta t$. You may take $\Delta t=0.5$.

\begin{aligned} \left.\frac{dy}{dx}\right|_{x_i}&\approx \frac{y(x_{i+1})-y(x_i)}{\Delta x}\qquad(2) \\ \left.\frac{d^2y}{dx^2}\right|_{x_i}&\approx \frac{y(x_{i+1})-2y(x_i)+y(x_{i-1})}{(\Delta x)^2}\qquad(3) \end{aligned}

In your Canvas announcements you will see that I have laid out a provisional plan for the remaining assessment.

We have now covered all the material, and my focus at this time is to support your learning through the remaining assessment.

The first assessment will take place 11:00 Tuesday 14 April 2020 and is based on Lab 6. The other assessment are provisionally set for 28 April (based on Lab 7), and 12 May (based on Lab 8). The following plan is designed around these.

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Easter Week 2 to Sunday 19 April

### 20% VBA Assessment Based on Lab 6

This test is worth 20% and will become available at 11:00 Tuesday 14 April, at which point you may download the student-number-individualised question paper.

More information under the VBA Test 2 (based on Lab 6) Practical Information announcement on Canvas.

If you have any further questions, I am answering emails seven days a week. Email before midnight Monday 13 April to be guaranteed a response early Tuesday 14 April. I cannot guarantee that I answer emails sent on Tuesday (although of course I will try).

### Catch Up/Revision

If prepared for (or have completed) VBA Test 2 (based on Lab 6), if you have not yet done so, you are advised to catch-up on the learning described in Week 9.

If you have already conducted this learning, and submitted a Lab 7 either back before 30 March, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback. If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 9.

Students can submit work — VBA or Theory, catch-up or revision — based on Week 9 to the Lab 7 VBA/Theory Catch-up/Revision I assignment on Canvas.

The remaining plans are provisional.

## Week 11

Similar to Easter Week 2.

## Week 12/13

### Catch Up/Revision

If you have not yet done so, you will undertake the learning described in Week 10.

If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 10.

If you have already conducted this learning, and submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Students will be able submit work — VBA or Theory, catch-up or revision — based on Week 10 to a Lab 8 VBA/Theory Catch-up/Revision assignment on Canvas.

I may have two due dates. Perhaps Sunday 3 May and Saturday 9 May.

## PROVISIONAL: Tuesday 12 May, Week 14: Assessment Based on Lab 8

In your Canvas announcements you will see that I have laid out a provisional plan for the remaining assessment.

We have now covered all the material, and my focus at this time is to support your learning through the remaining assessment.

If you have completed the learning tasks, and submitted Labs 6, 7, and 8, you should be in a good position to complete the remaining assessment.

However if this is not the case for you, it is not too late to get on top of all the material.

The first assessment will take place 11:00 Tuesday 14 April 2020 and is based on Lab 6. The other assessment are provisionally set for 28 April (based on Lab 7), and 12 May (based on Lab 8). The following plan is designed around these.

Especially if you have put in the hours over the last three weeks (inclusive), the demand on your time in terms of MATH7016 is now greatly diminished, but this plan also allows those who have not yet had a chance to engage in remote learning to begin.

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

### Final MCQ

If you have watched the Week 10 lectures, you should be able to do the final MCQ, MCQ VIII.

## Easter Week 1 to Sunday 12 April

### Catch Up

If you have not yet done so, undertake the learning described under Week 8 here. You may submit your Lab 6 to the UNGRADED Lab 6 Revision assignment on Canvas.

This ‘assignment’ has a due date of Easter Saturday. I have chosen this date so that you receive your feedback two days before the assessment.

### Revision

If you have already conducted this learning, and submitted a Lab 6 either back before 23 March, or after, I would invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

### Theory

If you feel like doing even more theory work on top of this, consider looking at the following exercises:

• p.59, Q. 1-3
• p.90, Q. 1-3

These can be submitted to UNGRADED Lab 6 Theory Exercises by Easter Saturday and will receive feedback Easter Sunday. If possible, submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

## Tuesday 14 April, Easter Week 2: Assessment Based on Lab 6

This test is worth 20% and will become available at 11:00 Tuesday 14 April, at which point you may download the student-number-individualised question paper.

You will be asked to write a program. The assessment runs from 11:00 to 12:30. The assessment is designed to be completed in 45 minutes, and there is a very generous 15 minutes allocated to submitting your work to Canvas. There is an additional 30 minutes grace period due to the remote nature of this assessment. Therefore you have until 12:30 to submit your work on Canvas. You can use the 90 minutes however you want but no submissions will be accepted after 12:30.

The remaining plans are provisional.

## Wednesday 15 April Easter Week 2 and Week 11/12 to Monday 27 April

### Catch Up

If you have not yet done so, you will undertake the learning described in Week 9. You will be able to submit your Lab 7 to an UNGRADED Lab 7 Revision assignment on Canvas.

I may have two due dates. Perhaps Sunday 19 April and Saturday 25 April.

### Revision

If you have already conducted this learning, and submitted a Lab 7 either back before 30 March, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

### Theory

If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 9.

## Wednesday 29 April Week 12 and Week 13/14 to Monday 11 May

### Catch Up

If you have not yet done so, you will undertake the learning described in Week 10. You will be able to submit your Lab 8 to an UNGRADED Lab 8 Revision assignment on Canvas.

I may have two due dates. Perhaps Sunday 3 May and Saturday 9 May.

### Revision

If you have already conducted this learning, and submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

### Theory

If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 10.

## PROVISIONAL: Tuesday 12 May, Week 14: Assessment Based on Lab 8

Please find a provisional plan for Weeks 10-13. I do not yet have any certainty around assessment. At this time it is my intention to deliver week-on-week up to 19 April, and then have review after this. Of course all plans are provisional in the current environment.

I want you to know from the perspective of MATH7016 that I am in daily communication with my Head of Department about a plan for your assessment. This has to be done properly, and indeed my HoD is helping me improve my plan.

Once the plan for the MATH7016 assessment is settled from my end, this will have to be integrated with all your other assessment.

This is why I have not been able to be definitive about remaining assessment. Trust however that as soon as concrete plans are in place these will be communicated to ye. There is no point in me speculating about assessment when all the ducks are not yet in a row.

My advice for the moment, for MATH7016 at least, is to focus at this time on learning rather than assessment.

This means my recommendation is, as far as MATH7016 is concerned, is to spend 7 hours a week on your learning.

Trust that whatever plan is devised for your MATH7016 assessment, it will put your needs first.

## Week 10 to Sunday 5 April

If you have completed the tasks outlined in the Week 9 summary, you can now begin on the below.

Any and all work, submitted at any time, will receive feedback.

### Lectures

There are about 82 minutes of lectures. You should schedule 2 hours and 10 minutes to watch them and take the notes in your manual. You need this extra time above 82 minutes because you will want to pause me.

Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.

### Lab 8

AFTER watching the lectures above, you should be able to do Lab 8 on p. 112.

Between now and Monday 6 April, you will be able to submit your work to Canvas, from which I will give you individual feedback. After 09:00 Tuesday 7 April I will download all student work and reply with feedback.

### MCQ

After watching the lectures above, you should be able to do the final MCQ, MCQ VIII.

You can still do MCQ7 if you have not yet done so. Submit your answers here before 10:00 Wednesday 1 April.

### Theory Exercises

You should be spending about 7 hours per week on MATH7016. If you have any time left over you should look at Theory exercises in the notes.

• p.111, Q. 1-2,
• p.113, Q. 4

If you still have not spent 7 hours I recommend looking back at:

• p.90, Q. 1-3

You can (carefully) take photos of your work (with questions labelled neatly) and submit to the Week 10 Theory Exercises those images on Canvas before midnight Sunday 5 April. After 09:00 Monday 6 April I will download all student work and reply with feedback.

If possible, submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

If you still haven’t spent 7 hours on MATH7016 maybe look at some other theory exercises in the manual. Any and all work will receive feedback.

Please find a provisional plan for Weeks 9-13. I do not yet have any certainty around assessment. At this time it is my intention to deliver week-on-week up to 19 April, and then have review after this. Of course all plans are provisional in the current environment.

## Week 9 to Sunday 29 March

If you have completed the tasks outlined in the Week 7 summary, you can now begin on the below.

### Lectures

There are about 80 minutes of lectures. You should schedule 2 hours and 15 minutes to watch them and take the notes in your manual. You need this extra time above 80 minutes because you will want to pause me.

Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.

### Lab 7

AFTER watching the first three lectures above, you should be able to do Lab 7 on p. 146.

Between now and Monday 30 March, you will be able to submit your work to Canvas, from which I will give you individual feedback. After 09:00 Tuesday 31 March I will download all student work and reply with feedback.

### MCQ

After watching the first three lectures above, you should be able to do MCQ VII.

You can still do MCQ VI if you have not yet done so. Submit your answers here before 10:00 Wednesday 25 March.

### Theory Exercises

You should be spending about 7 hours per week on MATH7016. If you have any time left over you should look at Theory exercises in the notes.

• p.104, Q. 1-2
• p.105, Q. 1

You can (carefully) take photos of your work (with questions labelled neatly) and submit to the Week 9 Theory Exercises those images on Canvas before midnight Sunday 29 March. After 09:00 Monday 30 March I will download all student work and reply with feedback.

If possible, submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

If you still haven’t spent 7 hours on MATH7016 maybe look at some other theory exercises in the manual. Any and all work will receive feedback.

## DME2C Lab 5: Runge-Kutta

DME2C are invited to do Lab 5: Runge-Kutta remotely.

I have set up an (ungraded) assignment on Canvas that you (DME2C student) can submit your Lab 5 work. After Tuesday 09:00 I will download all the submissions and give feedback on each.

Recall our overall framework for our programmes in MATH7016:

• Define variables
• Give initial values
• Loop that
• Prints current variables (of interest)
• Calculates next variables

For Runge-Kutta, as usual there will be an x and y variables, but also a number of $k_i$ variables which represent (estimates of) the slope at various points between the current and next value.

The loop must calculate the various $k_i$ BEFORE calculating the next and values. The next value is given as:

$y_{i+1}=y_i+h\cdot \phi(k_1,\dots,k_n)$,

where $\phi(k_1,\dots,k_n)$ is a weighted average of the slopes $k_i$. There is one for Euler’s Method (slope at previous), two for Heun’s (slope at previous and slope at predicted next), and we also look at ‘Common’ RK3 which uses three k variables, $k_1,\,k_2,\,k_3$, and ‘Classical’ (state-of-the-art) RK4 which uses four.

All of Lab 5 is to be done in VBA. Problem 4 is missing the formula:

$y_{i+1}=y_i+h\cdot \phi(k_1,\dots,k_4)$,

the relevant formula is on p. 74. Also the sign of the derivative is wrong (and I have the rocket fuel being ejected too quickly… use

$\displaystyle\frac{dv}{dm}=-\frac{5}{m}$.

## DME2C: Concept MCQ 5

I want to keep the (ungraded) MCQ league going — I have pledged €35 of my own cash for first (€20), second (€10), and third (€5), and it would be great to keep it going until the ‘end of the season’. The same three names have been leading the league for a few weeks so maybe they might falter now.

Well anyway, DME2C can enter MCQ5 by emailing me their selection (ABCDDB or whatever) before Tuesday 09:00.

## Week 8

Luckily enough from my point of view, because of St Patrick’s Day, I have already recorded lectures for next week.

First watch Goal Seek for Boundary Value Problems (less than 20 minutes).

Then you will be in a position to do Lab 6: Boundary Value Problems.

Between Tuesday 17 March and Monday 23 March, you will be able to submit your work to Canvas, from which I will give you individual feedback.

After this watch the rest of the first playlist, watch Intro to PDEs (less than 20 minutes), and then watch the Derivation of the Laplace Equation (40 minutes).

I will also send on an MCQ6 to keep the league going.

## Week 7

In the Tuesday 09:00 class we had Written Assessment 1.

The lab was based on Runge Kutta methods.

In the 12:00 lecture we did a (written) Shooting Method example.

In VBA we will also have MCQ V.

## VBA Assessment 1 – Week 6

DME2C have their VBA Assessment 1 Friday 09:00-11:00 and this will run 09:05-10:55

## Written Assessment 1 – Week 7

20 % Written Assessment, based on Weeks 1-5.

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

There are far more things I could examine.

Roughly, everything up to but not including Runge Kutta Methods (p.68).

Note the venue and time: Melbourn Hall, Tuesday 10 March, 09:30-10:30

## Week 6

After finishing talking about Runge-Kutta Methods, we looked at boundary value problems (in particular the Shooting Method).

In VBA we had VBA Assessment 1.

## MCQ League

Unless you are excelling, you are identified by the last five digits of your student number.

Please ask questions in the lab about questions you have gotten wrong. Students in red appear to not have a good handle on the material and should consider putting in extra time outside class in doing exercises (in the manuals).