CS计算机代考程序代写 scheme matlab algorithm 10/29/21, 12:17 AM Submit Assignment 3

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Assignment 3

Q1 Boundary Value Problem
3 Points

You are required to find the numerical solution to the following ordinary differential equation

in the domain . You are required to solve this equation using numerical methods

with the boundary conditions and .

Q1.1
1 Point

As shown in lectures, the derivative of a function can be calculated using the second

order central, and first order forward and backward difference schemes

−
dx2
d y2

−
dx

dy
2y = 10 cos(10x)

0 ≤ x ≤ 1
y(0) = 1 y(1) = 0

y(x)

(x ) ≈
dx

dy
i (y −i+1 y )/(2Δ)i−1

x(x ) ≈
dx

dy
0 (y −1 y )/Δ0

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The double derivative can be approximated as

Show that if you discretize equation shown in Q1 using the differentiation schemes above and

implementing the boundary conditions, you will get a set of equations that can be expressed as

where

(x ) ≈
dx

dy
N−1 (y −N−1 y )/Δ.N−2

(x ) ≈
dx2
d y2

i (y −i−1 2y +i y )/Δi+1
2

(x ) ≈
dx2
d y2

0 (y −0 2y +1 y )/Δ2 2

(x ) ≈
dx2
d y2

N−1 (y −N −1x 2y +N−2 y )/Δ .N−3
2

⎣⎢
⎢⎢⎢
⎢⎢⎢
⎢⎢⎢
⎢⎢⎡

1
α 1
0
0

⋮
0
0
0

0
β 1
α 2
0

⋮
0
0
0

0
γ 1
β 2
α 3

⋮
0
0
0

0
0
γ2
β 3

⋱
…
…
0

0
0
0
γ 3

⋱
α N−3

0
…

0
0
0
0

⋱
β N−3
α N−2

…
…
…
…

⋱
γ N−3
β N−2

0
0
0
0

⋮
0

γ N−2
1 ⎦

⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎤

⎩⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎨
⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎧ y 0

y 1
y 2
y 3

⋮
y N−3
y N−2
y N−1

⎭⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎬
⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎫

⎩⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎨
⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎧ a

Q 1
Q 2
Q 3

⋮
Q N−3
Q N−2
b

⎭⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎬
⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪
⎪⎪⎫

α =i +Δ2
1

2Δ
1

β =i − −Δ2
2

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where is the spacing between your equally spaced grid points. There are a

total of grid points. and . is the approximate value of .

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Q1.2
1 Point

Write a MATLAB program that uses the Thomas algorithm (from Assignment 2) to solve the

tridiagonal system found above in Q1.1. Compute your numerical solution using

and and plot vs for all values of .

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γ =i −Δ2
1

2Δ
1

Q =i 10 cos(10x )i

a = 1, b = 0

x =i+1 x +i Δ Δ
N x =0 0 x =N−1 1 y i y(x )i

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Δ = 0.2, 0.1
0.01 y i x i Δ

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Q1.3
0.5 Points

It is also possible to cast the problem shown in Q1 in terms of derivative matrices (using the

derivative formulae stated in Q1.1). Set up the first and second derivative matrices and use them

to construct the resulting matrix equations needed to solve the equation in Q1

and are your first and second derivative matrices respectively and is your

solution vector. This matrix should be “full” (with lots of zeros!) and you can use the

linsolve() function in MATLAB to solve the problem. Compute your numerical solution

using and and plot vs for all values of . Your answer should be

exactly the same as Q1.2. This will ensure that your derivative matrices are correct. Please

make sure you get this part right as you will be using the derivative matrices again in Q2 and

Q4 below.

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[D2] − [D1] − 2[I] {y} =[ ] {10 cos(10x)}.

[D1] [D2] {y}

Δ = 0.2, 0.1 0.01 y i x i Δ

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Q1.4
0.5 Points

Compare your approximated solutions from Q1.2 and Q1.3 to the exact solution to problem

shown in Q1

Comment on the accuracy of your numerical solutions.

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y(x) = −0.06982599398e +2x 1.166931859e −−x cos(10x) −
2626
255

sin(10x).
2626
25

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Q2
3 Points

This question is extracted from the book Numerical Analysis by Burden and Faires. The
governing equation for the deflection, , of a beam with two ends supported when

subjected to a uniform load of intensity (see figure above) is given by

where in is the length of the beam, lb/in is the modulus of elasticity,

in is the central moment of intertia, lb is the stress at the end points.

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y(x)
q

=
dx2
d y2

y +
EI

S
x(x −

2EI
q

l = 120 E = 3.0 × 107 2

I = 625 4 S = 1000

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Since the beam is supported at both ends, the boundary conditions are and

Q2.1
0.5 Points

Discretize the equation given in Q2 using the derivative schemes shown in Q1.1 and show that

the discretized equation can be expressed as a tridiagonal system (see Q1.1) with

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y(0) = 0 y(l) =
0

α =i Δ2
1

β =i − −Δ2
2

γ =i Δ2
1

Q =i x (x −2EI
q

i i l)

a = b = 0.

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Q2.2
1 Point

Write a computer program that uses the Thomas algorithm to solve the above set of tridiagonal

system for and lb/in. Approximate the deflection every 1.2 in, i.e. use

in. How can you be sure that your results are accurate? Do you need to use smaller

values of ? Justify your solution.

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Q2.3
0.5 Points

If you use Thomas algorithm to solve this system of equation, you need to derive the

expressions shown in Q2.1. One of the advantages of using derivative matrices (see Q1.3) is

that you can use them to solve the equation shown in Q2, without doing any further

mathematics. Use the derivative matrices in Q1.3 to construct the resulting matrix equations

needed to solve the equation in Question 2

q = 10, 15 25 y(x)
Δ = 1.2

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This matrix should be “full” (again with lots of zeros!) and you can use the linsolve() function

in MATLAB to solve the problem. Use the value that gives a solution that is “sufficiently

accurate” which you have justified in Q2.2.

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Q2.4
1 Point

The law requires that the maximum deflection is to be less than 1/300 in. Use the necessary

numerical techniques to find the range of the values of where the maximum deflection is

within the acceptable range. For this question, you can use your code either in Q2.2 or Q2.3.

and only the value which you justified in Q2.2.

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[D2] − (S/(EI)[I] {y} =[ ] {(q/(2EI))x(x − l)}.

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Q3 Initial Value Problem
5 Points

You are given that a very simple mathematical model of the motion of a sailboat in wavy waters

is given by the following ordinary differential equation

is the wind speed, is the angle in which the wind is blowing. is when the wind

is blowing in the positive direction (i.e. direction of travel of the sailboat) and is

when the wind is blowing directly opposite to the positive direction. is the amplitude

of the waves. Note that should be in radians when computing the term on the

right hand side of the equation given in this question above.

Q3.1
1 Point

Write a computer program to that uses the 2nd order Runge Kutta method to solve the equation

in Q3. Use this computer program to solve the equation for , and

. Assume that . Plot for . Use a small



100 =
dt2
d x2

40V cos(θ) +wind 5.5A sin −waves ( 10
x ) 50 .

V wind θ θ = 0o

x θ = 180o

x A waves
x/10 sin ( 10

x )

V =wind 12 θ = 30o

A =waves 2 dx/dt(0) = x(0) = 0 dx/dt t ∈ [0, 50]

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value of time step size so that you are reasonably sure that your results are accurate.

Justify the value of that you use in order to get an accurate solution.

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Q3.2
2 Points

Compute the solution for and m. Keep , . Plot

for for and m. According to this model, what happens to the

velocity, , of the sailboat when increases? Use the value of that you have

justified in Q3.1.

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Δt
Δt

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A =waves 0.1, 1 5 V =wind 12 θ = 30o dx/dt
0 ≤ t ≤ 50 A =waves 0.1, 1 5

dx/dt A waves Δt
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Q3.3
2 Points

Compute the solution for and . Keep , m.

Use a small value of so that you are reasonably sure that your results are accurate. You

should justify this value of . Plot for for all values of . According to

this model, what happens to the velocity of the sailboat when increases? Use the value of

that you have justified in Q3.1.

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Q4 Unsteady Convection–Diffusion
9 Points

The concentration of a contaminant as a function of location and time , , in an

environment where air is flowing at (constant) velocity and viscous diffusion can be

modelled by the following partial differential equation

θ = 0 , 10 , 30 , 50o o o o 90o V =wind 12 A =waves 5
h

Δt dx/dt 0 ≤ t ≤ 50 θ
θ Δt

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x t ϕ(x, t)
U ν

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Here the left hand side represents the convection of by the air and the right hand side the

diffusion of throughout the fluid in the pipe. You are interested in the value of for

during .

We’ll consider the case where the convection velocity has a constant value and the

diffusivity is and impose the following boundary condition

The initial concentration of the contaminant is given by a Gaussian function

Our aim in this question is to analyse the effects of various spatial discretisation schemes in

conducting the simulations and then find the location where the concentration of the

contaminant has decreased to 50% of its initial value.

Q4.1
0.5 Points

For Q4.1-Q4.6, we will be using the discretisation schemes shown in Q1.1. Explain how you can

use derivative matrices to write the discretise form of the convection diffusion equation as the

following set of ordinary differential equations

Show that for (assuming equally spaced grid points with spacing ),

+
∂t
∂ϕ

U =
∂x
∂ϕ

ν .
∂x2
∂ ϕ2

ϕ ϕ(x, t)
x ∈ [0, 20] t ∈ [0, 20]

U = 1
ν = 0.02

ϕ(x = 0, t) = 0

ϕ(x, 0) = e .−((x−2)/0.5)
2

{ϕ} =
dt

d
L {ϕ}.[ ]

N =x 6 Δ x

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and

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Q4.2
0.5 Points

You are given the boundary condition is , i.e. . Explain how you

would incorporate this boundary condition into in the equation above to get

{ϕ} =

⎩⎪⎪⎪⎪⎪
⎪⎪⎨
⎪⎪⎪⎪⎪
⎪⎪⎧ϕ 0
ϕ 1
ϕ 2
ϕ 3
ϕ 4
ϕ 5
⎭⎪⎪⎪⎪⎪
⎪⎪⎬
⎪⎪⎪⎪⎪
⎪⎪⎫

[L] = − +
2Δ x

⎣⎢
⎢⎢⎢
⎢⎢⎢
⎡−2

−1
0
0
0
0

2
0

−1
0
0
0

0
1
0

−1
0
0

0
0
1
0

−1
0

0
0
0
1
0

−2

0
0
0
0
1
2⎦
⎥⎥⎥
⎥⎥⎥⎥
⎤

Δ x2
ν

⎣⎢
⎢⎢⎢
⎢⎢⎢
⎡1

1
0
0
0
0

−2
−2
1
0
0
0

1
1

−2
1
0
0

0
0
1

−2
1
1

0
0
0
1

−2
−2

0
0
0
0
1
1⎦
⎥⎥⎥
⎥⎥⎥⎥
⎤

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ϕ(x = 0, t) = 0 ϕ =0 0
[L]

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Show that for ,

and

is the spacing between the (equally spaced) grid points.

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S A

{ϕ } =
dt

d ′ L {ϕ } +[ ′] ′ {P}.

N =x 6

{ϕ } =′

⎩⎪⎪⎪⎪⎪
⎨⎪
⎪⎪⎪⎪⎧ϕ 1ϕ 2
ϕ 3
ϕ 4
ϕ 5
⎭⎪⎪⎪⎪⎪
⎬⎪
⎪⎪⎪⎪⎫

[L ] =′ − +
2Δ x

⎣⎢
⎢⎢⎢⎢
⎡ 0

−1
0
0
0

1
0

−1
0
0

0
1
0

−1
0

0
0
1
0

−2

0
0
0
1
2⎦
⎥⎥⎥
⎥⎥⎤

Δ x
2

⎣⎢
⎢⎢⎢⎢
⎡−2

1
0
0
0

1
−2
1
0
0

0
1

−2
1
1

0
0
1

−2
−2

0
0
0
1
1⎦
⎥⎥⎥
⎥⎥⎤

{P} = ϕ

⎩⎪⎪⎪⎪⎪
⎨⎪
⎪⎪⎪⎪⎧ + 2Δ x

U
Δ x

2
ν

0
0
0
0

⎭⎪⎪⎪⎪⎪
⎬⎪
⎪⎪⎪⎪⎫

Δ x

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Q4.3
1 Point

Write a MATLAB program that solves the system of equation in Q4.2 above using the 2nd order

Runge–Kutta method. Carry out simulations with and .

Provide plots of the approximated solution at , , and (i.e. plot

, , , vs ). Does your solution look

reasonable for this value of ?

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Q4.4
1 Point

Plot all of the eigenvalues of the operator multiplied by and the stability region of the

2nd order Runge-Kutta method (see Lecture 18). You can use the eig() function in MATLAB to

obtain your eigenvalues for the operator . Investigate the effect of varying the size of the

N =x 40 Δt = 0.1
t = 0 t = 5 t = 10 t = 15

ϕ(x, t = 0) ϕ(x, t = 5) ϕ(x, t = 10) ϕ(x, t = 15) x
N x

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L′ Δt

L′

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time steps between and . For this question, use .

Discuss the effect of varying the time step on the stability of the system of equations. Can you

get a solution with ?

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Q4.5
1 Point

Using will not be able to give you a very good solution as the spatial resolution is not

sufficiently fine. Vary the number of points in your mesh using and

assume a constant time step of . Discuss the effect of varying on the stability of

the approximated solution. Can you get a solution using and ? Suggest

values of that you can use with to obtain a solution with and .

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Δt = 0.01, 0.05, 0.1, 0.2, 0.5 1 N =x 40

Δt = 1.0

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N =x 40
N =x 100, 200, 300, 500

Δt = 0.1 N x
N =x 500 Δt = 0.1

Δt N =x 100 N =x 500

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Q4.6
1 Point

Provide plots of the at , , and using and the

value that you have suggested in Q4.5.

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Q4.7
1 Point

So far, we have obtained approximated solutions of the partial differential equation shown in Q4

by calculating and using very basic (and not very accurate) 2nd order central

difference schemes (with 1st order approximation at the end nodes). As was shown in lectures,



ϕ(x, t) t = 0 t = 5 t = 10 t = 15 N =x 500 Δt

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∂ϕ/∂x ∂ ϕ/∂x2 2

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the accuracy of calculating derivatives and can be improved by using

spectral differentiation using Gauss-Lobatto points. For Q4.7-Q4.9, we will analyse the

properties of the numerical solution obtained using spectral differentiation and Gauss-Lobatto

points.

Using the lglnodes() and the DerivMatrix() MATLAB functions provided in Lecture 14,

construct derivative matrices that uses spectral Gauss-Lobatto scheme to calculate and

. Note that for this question, the derivative matrices (unlike the derivative matrices for

the 2nd order schemes) are ‘full’ matrices and do not contain any zeros.

Compute the solution using and and provide plots of the at

, , and . Compare with the solutions you obtain using finite difference

schemes in Q4.3. Which one do you think is more accurate?

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Q4.8
1 Point

∂ϕ/∂x ∂ ϕ/∂x2 2

∂ϕ/∂x
∂ ϕ/∂x2 2

N =x 40 Δt = 0.1 ϕ(x, t) t = 0
t = 5 t = 10 t = 15

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If you want to get a more accurate solution, you can increase . Perform stability analysis for

(i.e find the eigevalues of , similar to what you did in Q4.4 and Q4.5) and

suggest a value of that can be used to get a good solution.

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Q4.9
1 Point

Provide plots of the at , , and using and the

value that you have suggested in Q4.8.

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S A

N x
N =x 100 [L ]′

Δt

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Q4.10
1 Point

You are given that it safe for people to stand where the value of becomes less than 0.5.

Where would you ask these people to stand? To answer this question, you can use any of your

answer above and any other numerical method that feel is appropriate.

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Q5 Two Dimensional Boundary Value Problems
5 Points

In this question, you are asked to find the solution that satisfies the partial differential

equation below using the spectral collocation method (with Gauss Lobatto and also evenly

spaced points).

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u(x, y)

2 2

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In order to do that, you will need to first find the derivative matrices and use them to construct

the system of algebraic equation that looks something like

Q5.1 – Q5.2 will go through the steps on how you can set up the matrix . The solution is

obtained in Q5.4.

To extend the one-dimensional spectral collocation method with Gauss-Lobatto points into two

dimensions, we can tile the one dimensional mesh along the second direction. Suppose that

and are arrays containing th order ( point) one-dimensional meshes, then

we generate our two-dimensional mesh by repeating at a spacing determined by the

points in . The idea is akin to how MATLAB’s meshgrid() function works.

To illustrate this consider a two-dimensional mesh generated from the third order Gauss–

Lobatto points.

+
∂x2
∂ u2

=
∂y2
∂ u2

f(x, y)

[A]{u} = {f}

[A] {u}

{X} {Y } N N + 1
{X}

{Y }

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Here the nodes follow the lexicographic ordering from left to right, bottom to top.

The advantage of this type of construction is that it essentially decouples the axes, meaning

that we can take the one dimensional Lagrange basis functions and multiply them together to

produce the two-dimensional interpolants

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Hence the second derivatives are given by

For convenience, we write the coefficients in the vector

where .

Then the second derivative matrix may be written as

where is some operation on the one-dimensional second derivative matrix and the

identity matrix that produces the correct two-dimensional derivative matrix

Q5.1
1 Point

For the moment, assume a mesh. Show that the two dimensional derivative matrices are

given by

u(x, y) ≈ u L (x)L (y)
i=0

∑
N

j=0

∑
N

j,i i j

u (x, y)xx
u (x, y)yy

≈ u L (x)L (y)∑
i=0
N ∑

j=0
N

j,i i
′′

≈ u L (x)L (y).∑
i=0
N ∑

j=0
N

j,i i j
′′

u j,i

{u} = (u ,u , … ,u , … ,u ) =1 2 ℓ (N+1)2
T (u ,u , … ,u , … ,u )0,0 0,1 i,j N ,N

ℓ = 1 + i + (N + 1)j

u (x , y ) = [I] ⊗ [D ]{u}xx i i 2

u (x , y ) = [D ] ⊗ [I]{u}yy i i 2

⊗ [D ]2

[I]

2 × 2

2 2

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The result above can be extended to an mesh. The operation on two matrices is

called the Kronecker product which is implemented in MATLAB using the kron() function. For
an matrix and a matrix is defined as

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Q5.2
1 Point

Use the kron()} function in MATLAB to set up a matrix operator to calculate

[D ] 2 x

[D ] 2 y

⎝⎜
⎜⎜⎛

D 11
2

D 21
2

D 12
2

D 22
2

D 11
2

D 21
2

D 12
2

D 22
2 ⎠⎟
⎟⎟⎞

⎝⎜
⎜⎜⎛

D 11
2

D 21
2

D 11
2

D 21
2

D 12
2

D 22
2

D 12
2

D 22
2 ⎠⎟
⎟⎟⎞

N × N ⊗

m × n [A] p × q [B]

[A] ⊗ [B] = ⎝⎜⎜
⎛a [B]11

⋮
a [B]n1

⋯

⋱
⋯

a [B]1n

⋮
a [B].nn ⎠

⎟⎟⎞

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on a mesh of size . Test your matrix operator on the function

in the domain . You can use lglnodes() (to get the Gauss

Lobatto points) and the DerivMatrix() MATLAB functions provided in Lecture 14. To generate

the evenly spaced grid, use any inbuilt functions in MATLAB. Show that it works by using the

contour() or surf() function in MATLAB to plot the results of your matrix operation and

compare with the analytical function . You should show this for both

the evenly spaced grid and the Gauss Lobatto grid. You should get very good results, even

with relatively small values of .

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Q5.3
1 Point

Now we will apply the Spectral collocation method, and use the matrix operator above to solve

Poisson equation

+
∂x2
∂ u2

∂y2
∂ u2

N × N u(x, y) =
sin(πx) cos(πy) −1 < x, y < 1 −2π sin(πx) cos(πy)2 N No file chosenChoose Files  No file chosenChoose Files  2 2 No file chosenChoose Files No file chosenChoose Files No file chosenChoose Files No file chosenChoose Files 10/29/21, 12:17 AM Submit Assignment 3 | Gradescope https://www.gradescope.com/courses/278786/assignments/1367535/submissions/new 27/28 on the square with an arbitrary forcing function , which is also coupled with the dirichlet boundary condition for . Write a MATLAB function Poisson_solve(f,N,method)} that takes the inputs f , the forcing function, , the order of the mesh, and method , which is either the string GLo or Even for Gauss--Lobatto or Evenly spaced nodes respectively. Your function should output the the and coordinates as well as the solution at the start of this question in a form that can be directly inputted into the MATLAB functions contour() or surf() . Upload PDF file of your solution using link below Please select file(s) Select file(s) Upload MATLAB code using link below Please select file(s) Select file(s) Save Answer Q5.4 2 Points + ∂x2 ∂ u2 = ∂y2 ∂ u2 f(x, y) −1 < x, y < 1 f u = 0 x, y = ±1 N x y u   10/29/21, 12:17 AM Submit Assignment 3 | Gradescope https://www.gradescope.com/courses/278786/assignments/1367535/submissions/new 28/28 Using your function Poisson_solve() , plot the solution of the Poisson equation when Use both a Gauss--Lobatto and even spaced grid of order . Upload PDF file of your solution using link below Please select file(s) Select file(s) Upload MATLAB code using link below Please select file(s) Select file(s) Save Answer Save All Answers Submit & View Submission  f(x, y) = 10 sin {8x(y − 1)} 20   https://www.gradescope.com/courses/278786/assignments/1367535/submissions/95372409

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