COMP5930M - Scientific ComputingCoursework 2September 21, 2022Deadline09:00, Monday 19th DecemberTaskThe 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 numericalsolution.Matlab scripts referred to in this document can be downloaded from Minerva under LearningResources / Coursework. Matlab implementations of some algorithms have been provided aspart 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 stateddeadline. 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 anduse of that function.Standard late penalties apply for work submitted after the deadline.DisclaimerThis is intended as an individual piece of work and, while discussion of the work isencouraged, plagiarism of writing or code in any form is strictly prohibited.1. A one-dimensional PDE: Nonlinear parabolic equation[10 marks total]Consider the nonlinear parabolic PDE: find u(x, t) such that(1)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 withm 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 formulash2in space and the implicit Euler method in time.Define the nonlinear system F(U) = 0 that needs to be solved at each time step toobtain a numerical solution of the PDE (1).(b) Derive the exact expression for each of the non-zero elements in row i of the Jacobianfor this problem.(c) Solve the problem for different problem sizes: N = 81, 161, 321, 641, where N is thenumber of unknowns to solve for (choose the grid size m appropriately to get thecorrect N). Use NT = 40 time steps to solve from t0 = 0 to tf = 2. Create atable that shows the total computational time taken T , the total number of Newtoniterations 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 eachdifferent N :i. the total number of unknowns, Nii. the total computational time taken, Tiii. the total number of Newton iterations, Siv. the average time spent per Newton iteration, tS = T/SEstimate the algorithmic cost of one Newton iteration as a function of the number ofequations N from this simulation data. Does your observation match the theoreticalcost of the algorithms you have used?Hint: Take the values N and the measured tS(N) and fit a curve of the formtS = CNP by taking logarithms in both sides of the equation to arrive atlog tS = logC + P logN,then use polyfit( log(N), log(tS), 1 ) to find the P that best fits your data.(d) Repeat the timing experiment (c), but using now the Thomas algorithm sparseThomas.mto solve the linear system and the tridiagonal implementation of the numerical Ja-cobian computation tridiagonalJacobian.m. You will need to modify the code tocall newtonAlgorithm.m appropriately.Create a table that shows for each different N :i. the total number of unknowns, Nii. the total computational time taken, Tiii. the total number of Newton iterations, Siv. the average time spent per Newton iteration, tS = T/SEstimate the algorithmic cost of one Newton iteration as a function of the number ofequations N from this set of simulations. Does your observation match the theoreticalcost of the algorithms you have used?Hint: Take the values N and the measured tS(N) and fit a curve of the formtS = CNP by taking logarithms in both sides of the equation to arrive atlog 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[10 marks total]Consider the following PDE for u(x, y, z):= g(x, y, z), (2)defined on [x, y, z] ∈ [?10, 10]3 with u(x, y, z) = 0 on the boundary of the domain, andsome 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 hand n nodes in each coordinate direction, the nonlinear equation Fijk = 0 to be solved atan internal node may be written asCode for this problem can be downloaded from Minerva and is in the Q2/ folder. TheMatlab function runFDM.m controls the numerical simulation of the discrete nonlinearsystem (3). To solve the problem call runFDM(m), where m is the number of grid pointsin each direction so that the total number of unknowns is N = m3.(a) The Jacobian matrix is computed in the routine newtonAlgorithm. Extract theJacobian matrix J(x0) from the first Newton iteration of the algorithm. Then:i. Visualise the sparsity pattern of the Jacobian matrix using the command spyin the case m = 10 and include the plot in your report. How many non-zeroelements does it have?ii. Use the command lu to compute the LU-factorisation of A in the case m = 10.Visualise the sparsity pattern of the LU -factors using the command spy andinclude the plot in your report. What are the number of non-zero elements Land U , respectively?iii. Use these sparsity patterns to explain what we mean by fill-in of the LU -factors.Do you observe fill-in here?(b) Solve the tangent problem by using the iterative linear solver iterativeLinearSolve.mbased on the GMRES iteration (implemented by the gmres command in MATLAB).Use a fixed linear tolerance of linTol = 10?7 and 100 maximum linear iterations.Solve the problem for m = 10, 15, 20 and measure in each case the number of Newtoniterations and total number of linear iterations until convergence is achieved. Performthe same experiment with two different GMRES strategies:i. No preconditioner (default);ii. Inexact LU-preconditioner M ≈ LU computed by the MATLAB commandoptions.type = ’crout’;options.milu = ’row’;options.droptol = 0.05;[L,U] = ilu(A,options);M=L*U;(modify the file iterativeLinearSolve.m to implement the preconditioner).Create a table that measures the total number of Newton iterations and total numberof linear iterations in each case for m = 10, 15, 20 for both choices of preconditioningstrategies. How does preconditioning affect the number of linear iterations required?(c) Modify the file iterativeLinearSolve.m to implement an inexact Newton-Krylovmethod where the GMRES linear stopping tolerance is chosen aslinTol = max{η ‖F(xk)‖, 10?7},where F(xk) is the nonlinear residual at iteration k and the parameter η is chosenfrom the Eisenstat-Walker rule:η = 0.1‖F(xk)‖2‖F(xk?1)‖2 .Repeat the experiment from (b) and update the table to include the number ofNewton iterations and total number of linear iterations in this case. How does theinexact Newton-Krylov influence the total number of iterations required in this case?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 schemeThis piece of work is worth 20% of the final module grade.There are 20 marks in total.1. One-dimensional PDE [10 marks total](a) Formulation of the discrete problem [3 marks](b) Jacobian structure [3 marks](c) Timings and analysis of complexity, general case [2 marks](d) Timings and analysis of complexity, tridiagonal case [2 marks]2. Three-dimensional PDE [10 marks total](a) Analysis of the sparsity of the Jacobian matrix and the the LU factors [4 marks](b) Experiments with the iterative GMRES linear solver [2 marks](c) Experiments with the inexact Newton-Krylov method [4 marks]