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QUESTION 4 [40 MARKS TOTAL] Background In this question, you will be solving the 2D heat equation defined in Eqn 3 in a rectangular domain of x 1 and -0.5 s y 505 The domain has an initial temperature of zero and the boundary conditions are given by T(x.y)=exp(1-x-y2)sin(x+y²) Eqn. 4 You are asked to discretize the rectangular domain into Nx grid points in the x-direction and Ny grid points in the y-direction. Note that the first and last points in both the x- and y-directions are boundary conditions, which means that there will be a total of (Nx-2)x (Ny 2) unknowns Q4a in your PDF for Question 1. outline the algorithm needed to solve Eqn 3 using the BTCS scheme from time = o to time t., with a constant timestep, At (assume that to NtAt). Include the major steps required to set the problem up (don't include minor steps like setting grid spacing etc.) This algorithm must be in the form of pseudo code or a flow chart. Example of a pseudo code for solving an ODE using the Euler method is shown in the smippet below. Set up parameters of the problem Determine stepsize and number of steps for each time step Solve the Euler method Store calculated values in vector end Q4b Write a Matlab function that sets up the Laplacian matrix L and the boundary condition vector BC Your function header must be written as function BC) - Mat Set up (Nx. Ny. xg. yg, alpha, acf) where Nx and Ny are the number of points in the x- and y-directions, respectively . xg and Y8 are column vectors containing the grid points of , and y, respectively. alpha is a, i.e. the thermal diffusion coefficient BCF is function describing the boundary condition (see Eqn 4) Lis the [(Nx-2)*(Ny-2) (Nx 2)*(Ny-2)] Laplacian matrix BC is column vector on the RHS that accounts for the boundary conditions NOTE: You can define an anonymous function that converts a 2D node number (j.k) = k) to a global node number for the unknowns. Your function will look like nn=@(j.k.Nx) {some function of k, Nx): (See Workshops 25-27 for examples of what this looks like). When setting up the matrix problem the function no (or at least a similar process) can be used to reliably determine the row and column number of each entry in the Laplacian matrix and the row number in the BC matrix Q4c Modily Lab06_Q4.m to salve the 2D Heat equation using the BTCS scheme (i.e. Implement your algorithm from Q4a) together with the function you have written in Q4b. Set a 0.1 and time step At 1s, and solve the PDE from t 0 to =50s for (Na, Ny) (21, 11), [41, 21), (61, 31) and (81, 41). Using the command 'tic' and 'toc', determine how much time is spent setting the matrix problem up. i.e. the time spent assembling the Laplacian matrix versus how much time is spent undertaking the time-stepping solution. (NOTE: The time taken for plotting should not be included as part of the time taken for time-stepping). Print out the times for each (Nx, Ny) grid and write . few sentences comparing the times and explain what they mean. In separate figures, plot contours of your salution Including the boundary values for each (Nx, Ny). You will have to reshape your 10 vector of unknowns into a 20 matrix and embed that into a slightly bigger 20 matrix and add boundary conditions. Q4d Repeat Q4c with timestep of At - 50: $ (Just do this inside the same script file (Lab06_Q4.m) with a loop for the 2 different time steps). Is the solution stable or unstable in this case? Should it be? Write a comment to the command window that compares your solution calculated with one large time step to that obtained in 50 smaller time steps. NOTE: the following MATLAB code will contour a 20 grid of data values stored in Oplot, where xg is a 1-D vector that specifies the grid x-locations (length=Nx) and Yg is a 1-D vector that specifies the grid locations [length=Ny) figure clev-linspace(0.25,1.251: Set the desired contour levels

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% Applies BTCS algorithm for given parameters and returns:
%   1. the solution in matrix form (QnMax) with added boundary conditions
%   2. grid points on x and y axis (xg, yg)
%   3. total times needed to form Laplace matrix and to solve linear system
%   (mat_time, sol_time)
function [QnMat, xg, yg, mat_time, sol_time] = BTCS(alpha, dt, t0, tmax, Nx, Ny)
    xg = -1:2/(Nx-1):1;
    yg = -0.5:1/(Ny-1):0.5;

    BCf = @(x,y) exp(1-x^2-y^2)*sin(x^2+y^2);

    nL = (Nx-2)*(Ny-2);

    % Starting values are all zeros, outside boundaries
    Qn = zeros(nL,1);

    % matrix formation time and linear system solution time
    mat_time = 0;
    sol_time = 0;
    for t=t0+dt:dt:tmax
       % form the Laplace matrix and its RHS side from boundary conditions
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