代做program、代写Python编程设计

日期：2023-12-18 08:40

1. Assignment guidance

In this coursework, you will use software libraries to explore di?erent numerical schemes

and you will analyse the methods and results.

2. Assessment tasks

In this coursework, you will be writing a report for your boss as BigNumComp Inc

helping them to choose a numerical method. You should assume she has the knowledge

of second year undergraduate Computer Science student at the University of Leeds.

Problem

Your boss wants to find good numerical methods for solving predator-prey models.

Predator-prey models describe the evolution of two di?erent co-existing species: one

is the predator and one is the prey.

The problem can be described through the following system of di?erential equations

for x(t), the prey, and y(t) the predator:

dx

dt = ?x  xy + f(t)

dy

dt = xy  y + g(t),

where ?, , ,  are positive real numbers and f and g are given functions of t which

relate to external migration into or out of the system.

Your boss, helpfully, suggests two test cases for you to consider:

(a) Use the parameter set ? =  =  =  = 1,

f(t) =  sin (t)  (cos (t))2  cos (t) g(t) = sin (t) + (cos (t))2  cos (t),

and initial condition x(t = 0) = 2 and y(t = 0) = 0, which has exact solution

x(t) = 1 + cos(t) y(t)=1  cos(t).

(b) Use the parameter set ? = 2/3,  = 4/3,  = 1,  = 1, f(t) = 0, g(t) = 0 and

x(t = 0) = 0.9, y(t = 0) = 0.9. For this case there is no nice form for the exact

solution but your boss tells you that the solution should repeat itself and the

maxima of each population should be consistent (i.e. not change) over time.

You are tasking with finding the best solver in terms of accuracy and eciency. You

should write a report (following the template laid out below) evidencing your recommendations and reasoning.

Task

You recognise this problem as being solvable as a system of di?erential equations. You

have seen some methods in the course but know there are better methods available. In

2

your preliminary research you find a company internal solver for systems of di?erential

equations (attached as solvers.py). You will solve and analyse the results of solving

the predator prey model for two di?erent methods within solvers.py and one method

from the module which you think is the most appropriate for this problem.

You should include all the sections in the template in your report. There is guidance of

what to include in each section and a guidance word limit for each section too. Code,

tables and equations do not count in the word limit. The word limit is only guidance

and no penalties will be introduced for going over the limit. You should aim to write

less than the word count to ensure your writing is concise and understandable to your

audience.

You should submit a jupyter notebook including all computations as your report.

You should write in full sentences throughout to guide the reader through what you

are doing. There is no need to include the file solvers.py in your solution. You

should write text in Markdown blocks and include all code for the implementation and

generating results in Code blocks.

Library documentation

solvers Solve di?erential equations.

Solve the di?erential equation(s) specified by

y0

(t) = f(t, y) subject to y(t0) = y0.

The problem is solved using a specified method from t0 to a final time T using a time

step dt.

Parameters

rhs A python function describing the right hand side function f of the di?erential

equation. The function takes two arguments: the first represents t for time and

the second represents the solution y which may be either a floating point type or

a numpy array for the case of multiple di?erential equations.

y0 The starting value of y - accepts either a floating point type or a numpy array for

the case of multiple di?erential equations.

t0 The starting time t0.

dt The time step dt.

T The final or stopping time.

method The method used to advance the solver given as a string. The method should

be one of

? "Heun"

? "Ralston"

? "Van De Houwen"

3

? "SSPRK3"

? "Runge-Kutta"

? "3/8-rule"

? "Ralston-4".

Returns

t The time points where the solution was found as a list

y The estimate of the solution at each time point as a list

Sample code

Download the file solver.py and place it in the same folder as your code.

An example for solving the a single di?erential equation:

from solvers import solver

def rhs (t, y) :

return -y

y0 = 1.0

t0 = 0.0

dt = 0.1

T = 1.0

t, y = solver (rhs , y0 , t0 , dt , T)

An example for solving the a system of di?erential equation:

import numpy as np

from solvers import solver

def rhs (t, y) :

return np. array ([-y[1], y [0]])

y0 = np. array ([1.0, 0.0])

t0 = 0.0

dt = 0.1

T = 1.0

t, y = solver (rhs , y0 , t0 , dt , T)

Solution template

4

(a) Implementation. Write code to be able to run three methods you have chosen for

arbitrary initial conditions (x(t = 0), y(t = 0)), time step (dt), model parameters

(?, , , ) and functions f and g. [50 words]

(b) Results. Simulate and show results for each of the test cases suggested by your

boss for a range of time steps until a final time T. For test case 2a, you should use

T = 2.5? and (at least) dt = T/100,T/200,T/400,T/800,T/1600. For test case

2b, you should use T = 30 and (at least) dt = T/100,T/200,T/400,T/800,T/1600.

You should demonstrate how solutions look for each method, and the accuracy

and eciency of each approach. [50 words]

(c) Analysis. Comment on the eciency and accuracy of each approach. [250 words]

(d) Conclusion. Compare the methods that you have results for, and any other relevant methods from the module, and make a recommendation of which method

you think is best. [300 words]

3. General guidance and study support

Examples of how to approach each aspect of this coursework is given in lectures and

using the online notes.

You may wish to read more about predator prey models (also known as Lotka Volterra

equations) to help with your analysis:

? Wikipedia: Lotka Volterra equations

? Mathworld: Lotka Volterra equations

Further support for this assessment is given through the MS Class Team. Details of

further support sessions will be given closer to the deadline.

4. Assessment criteria and marking process

Your work will be assessed on your code implementation, your results and their presentation, your analysis of the method and results, and your writing quality. Work will

be marked as a final assessment for this module so your mark will only be given back

as part of your final grade.

5. Presentation and referencing

The quality of written English will be assessed in this work - further details in the

Rubric below. As a minimum, you must ensure:

? Paragraphs are used

? There are links between and within paragraphs although these may be ine?ective

at times

? There are (at least) attempts at referencing

5

? Word choice and grammar do not seriously undermine the meaning and comprehensibility of the argument

? Word choice and grammar are generally appropriate to an academic text

These are pass/ fail criteria. So irrespective of marks awarded elsewhere, if you do not

meet these criteria you will fail overall.

6. Submission requirements

Please submit your work via Gradescope by the deadline given. You should submit a

jupyter notebook with all your code, text and results included in a single document.

You are recommended to “reset the kernel” and “Run all cells” again before you submit.

7. Academic misconduct and plagiarism

? Leeds students are part of an academic community that shares ideas and develops

new ones.

? You need to learn how to work with others, how to interpret and present other

people’s ideas, and how to produce your own independent academic work. It is

essential that you can distinguish between other people’s work and your own, and

correctly acknowledge other people’s work.

? All students new to the University are expected to complete an online Academic

Integrity tutorial and test, and all Leeds students should ensure that they are

aware of the principles of Academic integrity.

? When you submit work for assessment it is expected that it will meet the University’s academic integrity standards.

? If you do not understand what these standards are, or how they apply to your

work, then please ask the module teaching sta? for further guidance.

By submitting this assignment you are confirming that the work is a true

expression of your own work and ideas and that you have given credit to

others where their work has contributed to yours.

8. Assessment/marking criteria grid

The final assessment will be marked out of 50 according the following rubric.

Algorithm implementation (10 marks)

6

Marks Description

9-10 Algorithm(s) implemented accurately and eciently.

Professional quality code (Uniform formatting, unit tests

where appropriate). No eciency problems. Informative

comments.

7-8 Algorithm(s) implemented with no errors. Some

eciency problems. Helpful comments throughout.

6-7 Algorithm(s) implemented with no errors. Some helpful

comments.

5-6 Algorithm(s) implemented with minor errors. Some

comments.

0-4 Serious issues with code implementation resulting in

inaccurate results.

Presentation of results (15 marks)

Marks Description

13-15 Results in a variety of appropriate formats (i.e. tables,

plots, etc). Results and extensive additional useful

information shown. Plots and tables labelled accurately.

10-12 Results in a variety of appropriate formats (i.e. tables,

plots, etc). Results and additional useful information

shown. Plots and tables labelled accurately.

7-9 Results in a variety of appropriate formats (i.e. tables,

plots, etc). Results accurately shown.

4-6 Attempts are carefully formatting results suitable for

technical audience. Results accurately shown.

0-3 Basic or very limited results shared.

7

Analysis of results (20 marks)

Marks Description

17-20 Critical explanations using additional computational

experiments and making reference to appropriate

external literature covering all methods

13-16 Critical explanations using additional computational

experiments covering all methods

9-12 Descriptive explanations using further analysis of

suggested computational experiments covering all

methods

5-8 Descriptive explanations based purely on suggested

experiments covering all methods

0-4 No or very limited results explained

Writing (5 marks)

Marks Description

5 Outstanding structure and clarity of writing, all in a

suitable language. No errors.

4 Clear structure and writing in suitable language. Some

minor errors.

3 Well structured with mostly clear writing in suitable

language. Some errors.

2 Structure could have been improved. Some text required

careful reading. Language not appropriate for technical

report.

8

Marks Description

1 Poor presentation and structure with unclear or

confusion descriptions. Many errors.

A External library reference

The source code for solvers.py:

from typing import Callable , List , Tuple , TypeVar

import numpy as np

# Butcher tables for each of the methods used

TABLEAU = {

" Heun ": (

np. array ([[0.0, 0.0], [1.0, 0.0]]),

np. array ([0.5, 0.5]),

np. array ([0.0, 1.0]),

) ,

" Ralston ": (

np. array ([[0.0, 0.0], [2 / 3, 0.0]]),

np. array ([0.25, 0.75]),

np. array ([0.0, 2 / 3]),

) ,

" Van der Houwen ": (

np. array ([[0.0, 0.0, 0.0], [1 / 2, 0.0, 0.0], [0.0,

0.75, 0.0]]),

np. array ([2 / 9, 1 / 3, 4 / 9]),

np. array ([0.0, 1 / 2, 3 / 4]),

) ,

" SSPRK3 ": (

np. array ([[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.25,

0.25, 0.0]]),

np. array ([1 / 6, 1 / 6, 2 / 3]),

np. array ([0.0, 1.0, 1 / 2]),

) ,

"Runge - Kutta ": (

np. array (

[

[0.0, 0.0, 0.0, 0.0],

9

[0.5, 0.0, 0.0, 0.0],

[0.0, 0.5, 0.0, 0.0],

[0.0, 0.0, 1.0, 0.0],

]

) ,

np. array ([1 / 6, 1 / 3, 1 / 3, 1 / 6]),

np. array ([0.0, 0.5, 0.5, 1.0]),

) ,

"3/8- rule ": (

np. array (

[

[0.0, 0.0, 0.0, 0.0],

[1 / 3, 0.0, 0.0, 0.0],

[-1 / 3, 1.0, 0.0, 0.0],

[1.0, -1.0, 1.0, 0.0],

]

) ,

np. array ([1 / 8, 3 / 8, 3 / 8, 1 / 8]),

np. array ([0.0, 1 / 3, 2 / 3, 1]),

) ,

" Ralston -4": (

np. array (

[

[0.0, 0.0, 0.0, 0.0],

[0.4, 0.0, 0.0, 0.0],

[0.29697761 , 0.15875964 , 0.0, 0.0],

[0.21810040 , -3.05096516, 3.83286476, 0.0],

]

) ,

np. array ([0.17476028, -0.55148066, 1.20553560,

0.17118478]),

np. array ([0.0, 0.4, 0.45573725 , 1.0]),

) ,

}

# types for y variable in solver

y_type = TypeVar (" y_type ", np. ndarray , np. double )

def solver (

rhs : Callable [[ np. double , y_type ] , y_type ] ,

y0: y_type ,

10

t0: np. double ,

dt: np. double ,

T: np. double ,

method : str ,

) -> Tuple [ List [np. double ] , List [ y_type ]]:

"""

Solve the differential equation (s).

Solve the differential equation specified by

y ’(t) = rhs (t, y) subject to y( t_0 ) = y_0 .

The problem is solved numerical using METHOD from t0 to T

using a time step dt.

Parameters

----------

rhs

A function describing the right hand side of the

differential equation (s)

y0

The starting value of y

t0

The starting value of t

dt

The time step

T

The final or stopping time

method

The method used to advance to solver . method

should be one of:

Heun , Ralston , Van De Houwen , SSPRK3 , Runge -Kutta

, 3/8- rule , Ralston -4

Returns

-------

t

The time points where the solution was found

y

11

The estimate of the solution at each time point

"""

# set initial data into solution arrays

t_out = [t0]

y_out = [y0]

# extract method helpers

matrix , weights , nodes = TABLEAU [ method ]

s = len ( weights )

k: List [ y_type | None ]=[ None for _ in range (s) ]

# count steps

timesteps = int (T / dt)

# time loop

for step in range ( timesteps ):

# build k’s

for i in range (s):

temp = sum ( matrix [i, j] * k[j] for j in range (i) )

k[i] = rhs ( t_out [-1] + dt * nodes [i] , y_out [-1] +

dt * temp )

y_update = sum ([k[i] * weights [i] for i in range (s)])

y_new = y_out [-1] + dt * y_update

t_new = t_out [-1] + dt

t_out . append ( t_new )

y_out . append ( y_new )

return t_out , y_out

def example_code_1 ():

"""

Example code for single differential equation

The problem is y ’(t) = y subject to y (0) = 1.0.

The problem is solved with dt = 0.1 until T = 1.0 using

Heun ’s method

12

"""

def rhs1 (t: np. double , y: np. double ) -> np. double :

return -y

t, y = solver (rhs1 , 1.0, 0.0, 0.1, 1.0, " Heun ")

def example_code_2 ():

"""

example code for system of differential equations

The problem is (x ’(t), y ’(t)) = ( -y(t), x(t)) subject to

(x (0) , y (0) ) = (1.0, 0.0)

The problem is solved with dt = 0.1 until T = 1.0 using

the Runge - Kutta method

"""

def rhs2 (t: np. double , y: np. ndarray ) -> np. ndarray :

return np. array ([-y[1], y [0]])

t, y = solver (rhs2 , np. array ([1.0, 0.0]), 0.0, 0.1, 1.0,

"Runge - Kutta ")

if __name__ == " __main__ ":

for method , ( matrix , weights , nodes ) in TABLEAU . items ():

# test methods are explicit

np. testing . assert_almost_equal (np. tril ( matrix ) ,

matrix )

# test methods are consistent

np. testing . assert_almost_equal ( sum ( weights ), 1.0)

# test dimensions match

n, m = matrix . shape

assert n == m

assert n == len ( weights )

assert n == len ( nodes )

example_code_1 ()

example_code_2 ()

13