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COMP5930M - Scientific Computing

Coursework 2

November 16, 2021

Deadline

10:00, Monday 13th December

Task

The numbered sections of this document describe problems which are modelled by partial differ-

ential equations. A numerical model is specified which leads to a nonlinear system of equations.

You will use one or more of the algorithms we have covered in the module to produce a numerical

solution.

Matlab scripts referred to in this document can be downloaded from Minerva under Learning

Resources / Coursework. Matlab implementations of some algorithms have been provided as

part of the module but you can implement your solutions in any other language if you prefer.

You should submit your answers as a single PDF document via Minerva before the stated

deadline. MATLAB code submitted as part of your answers should be included in the document.

MATLAB functions should include appropriate help information that describe the purpose and

use of that function.

Standard late penalties apply for work submitted after the deadline.

Disclaimer

This is intended as an individual piece of work and, while discussion of the work is

encouraged, plagiarism of writing or code in any form is strictly prohibited.

1. A one-dimensional PDE: Nonlinear parabolic equation

[20 marks total]

Consider the nonlinear parabolic PDE: find u(x, t) such that

in the spatial interval x ∈ (0, 1) and time domain t > 0.

Here and α are known, positive, constants.

Boundary conditions are specified as u(0) = 0 and u(1) = 1.

Initial conditions are specified at t = 0 as u(x, 0) = x.

We numerically approximate (1) using the method of lines on a uniform spatial grid with

m nodes on the interval [0, 1] with grid spacing h, and a fixed time step of ?t.

Code for this problem can be downloaded from Minerva and is in the Q1/ folder.

(a) Find the fully discrete formulation for (1) using the central finite difference formulas

h2

in space and the implicit Euler method in time.

Define the nonlinear system F(U) = 0 that needs to be solved at each time step to

obtain a numerical solution of the PDE (1).

(b) Solve the problem for different problem sizes: N = 81, 161, 321, 641 (choose m ap-

propriately for each case). Use NT = 40 time steps. Create a table that shows the

total computational time taken T , the total number of Newton iterations S, and the

average number of Newton iterations per time step TS = T/S. Time the execution

of your code using Matlab functions tic and toc. Show for each different N :

i. the total number of unknowns, N

ii. the total computational time taken, T

iii. the total number of Newton iterations, S

iv. the average time spent per Newton iteration, tS = T/S

Estimate the algorithmic cost of one Newton iteration as a function of the number of

equations N from this simulation data. Does your observation match the theoretical

cost of the algorithms you have used?

Hint: Take the values N and the measured tS(N) and fit a curve of the form

tS = CN

P by taking logarithms in both sides of the equation to arrive at

log tS = logC + P logN,

then use polyfit( log(N), log(tS), 1 ) to find the P that best fits your data.

(c) Derive the exact expression for row i of the Jacobian for this problem.

How many nonzero elements are on each row of the Jacobian matrix?

(d) Repeat the timing experiment (c), but using now the Thomas algorithm sparseThomas.m

to solve the linear system and the tridiagonal implementation of the numerical Ja-

cobian computation tridiagonalJacobian.m. You will need to modify the code to

call newtonAlgorithm.m appropriately.

Create a table that shows for each different N :

i. the total number of unknowns, N

ii. the total computational time taken, T

iii. the total number of Newton iterations, S

iv. the average time spent per Newton iteration, tS = T/S

Estimate the algorithmic cost of one Newton iteration as a function of the number of

equations N from this set of simulations. Does your observation match the theoretical

cost of the algorithms you have used?

Hint: Take the values N and the measured tS(N) and fit a curve of the form

tS = CN

P by taking logarithms in both sides of the equation to arrive at

log tS = logC + P logN,

then use polyfit( log(N), log(tS), 1 ) to find the P that best fits your data.

2. A three-dimensional PDE: Nonlinear diffusion

[20 marks total]

Consider the following PDE for u(x, y, z):

defined on [x, y, z] ∈ [?10, 10]3 with u(x, y, z) = 0 on the boundary of the domain, and

some known function g(x, y, z) that does not depend on u.

Using a finite difference approximation to the PDE (2) on a uniform grid, with grid size h

and n nodes in each coordinate direction, the nonlinear equation Fijk = 0 to be solved at

an internal node may be written as

Code for this problem can be downloaded from Minerva and is in the Q2/ folder. The

Matlab function runFDM.m controls the numerical simulation of the discrete nonlinear

system (3).

(a) Run the model using runFDM.m with grid dimension m = 5, 8 and 11 for tol = 10?8

and initial guess u(x, 0) = 0 for all x. Create a table that shows for each different N :

i. the total number of unknowns, N

ii. the total computational time taken, T

iii. the total number of Newton iterations, S

iv. the average time spent per Newton iteration, tS = T/S

Estimate the algorithmic cost of a single Newton iteration in terms of the size of

the nonlinear system N , i.e. T = O(Np). How does it compare to the theoretical

worst-case complexity?

Hint: Take the values N and the measured tS(N) and fit a curve of the form tS =

CNP by taking logarithms in both sides of the equation to arrive at

log tS = logC + P logN,

then use polyfit( log(N), log(tS), 1 ) to find the P that best fits your data.

(b) Replace in runFDM.m the call to a dense Jacobian implementation fnJacobian.m with

a call to a sparse implementation, buildJacobian.m. You will need to uncomment

the following function calls before calling the Newton algorithm:

[ A, nz ] = sparsePattern3DFD( m ); % 3d FDM sparsity pattern

[ loc, nf ] = sparseJac( A,N ); % function calls required to evaluate J

i. Visualise the sparsity pattern of A using the command spy in the case m = 8

and include the plot in your report. Note: The matrix A has the same sparsity

pattern as the Jacobian for this problem.

ii. Use the command lu to compute the LU-factorisation of A in the case m = 8.

What are the number of non-zero elements L and U , respectively?

iii. Visualise the sparsity pattern of the LU -factors using the command spy and

include the plot in your report. Can you observe something unexpected about

the matrix L? How can you explain this perceived discrepancy?

iv. Use these matrices as an example to explain what we mean by fill-in of the

LU -factors.

(c) Use the MATLAB code provided to assemble the Jacobian matrix J for problem (2)

with the grid size m = 8, and consider solving the linear problem Jx = b, where

b = [1, 1, 1, . . . , 1]T . Solve this linear problem using three different solution methods:

i. LU-decomposition direct method (directLU.m);

ii. Jacobi iterative method (jacobi.m) using the initial guess x0 = b and tol = 10

6;

iii. Gauss-Seidel iterative method (gaussseidel.m) using the initial guess x0 = b

and tol = 10?6.

Solve the problem 100 times with each method and report the total time used for

each method (hint: use the commands tic and toc), plus for the iterative methods

the number of iterations taken.

(d) Replace in runFDM.m the call to linearSolve.m function that uses dense linear alge-

bra with the alternative matlabLinearSolve.m that uses sparse linear algebra. Solve

the problem using the sparse implementation for m = 15, 19 and 23. Create a table

that shows for each different N :

i. the total number of unknowns, N

ii. the total computational time taken, T

iii. the total number of Newton iterations, S

iv. the average time spent per Newton iteration, tS = T/S

Estimate the algorithmic cost of this new sparsity-based implementation using the

timings of the numerical experiments. Compare the results with those of 2(a).

Hint: Take the values N and the measured tS(N) and fit a curve of the form tS =

CNP by taking logarithms in both sides of the equation to arrive at

log tS = logC + P logN,

then use polyfit( log(N), log(tS), 1 ) to find the P that best fits your data.

Learning objectives

Formulation of sparse nonlinear systems of equations from discretised PDE models.

Measuring efficiency of algorithms for large nonlinear systems.

Efficient implementation for sparse nonlinear systems.

Mark scheme

This piece of work is worth 20% of the final module grade.

There are 40 marks in total.

1. 1d PDE [20 marks total]

(a) Formulation of the discrete problem [5 marks]

(b) Timings and analysis of complexity, general case [5 marks]

(c) Jacobian structure [5 marks]

(d) Timings and analysis of complexity, tridiagonal case [5 marks]

2. 3d PDE [20 marks total]

(a) Simulations with dense version of the solver [5 marks]

(b) Analysis of the sparsity of the Jacobian matrix and the the LU factors [5 marks]

(c) Comparison of linear solvers [5 marks]

(d) Simulations with sparse version of the solver [5 marks]

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