Create A Program to Create Computational Methods in Matlab Assignment Solution.

function [t,u] = CentredFiniteDifference(Nx, tend, v0, v, u0)

    % Solve

    nsteps = 4*Nx;

    dt = tend/double(nsteps);

    dx = (2*pi)/Nx;

    u = zeros(length(u0),nsteps+1);

    u(:,1) = u0;

    for i=1:nsteps

        u(:,i+1) = u(:,i) + dt*centred_f(u(:,i), v0, v, dx);

    end

    t = 0:dt:tend;

end

PSEUDO SPECTRAL

function [t,u] = PseudoSpectral(Nx, tend, v0, v, u0)

    % Solve

    nsteps = 4*Nx;

    dt = tend/double(nsteps);

    u = zeros(length(u0),nsteps+1);

    u(:,1) = u0;

    for i=1:nsteps

        u(:,i+1) = u(:,i) + dt*spectral_f(u(:,i), v0, v);

    end

    t = 0:dt:tend;

end

clc, clear all, close all

global v v0

%% Task 1, Part 2

v = 0.1;

Nx = 64;

v0 = 0.5;

x = linspace(0, 2*pi, Nx);

t = linspace(0, 2, 4*Nx);

%% First, solve with u(0) = cos(3x)

u0 = cos(3*x);

[t,y1] = ode45(@(t,u)general_ivp(t,u), t, u0);

%% Now, solve with u(0) = cos(7x)

u0 = cos(7*x);

[t,y2] = ode45(@(t,u)general_ivp(t,u), t, u0);

%% Plot

figure

subplot(1,2,1)

plot(t,y1), grid on

xlabel('Time (s)');

ylabel('{u(x,t)}');

title('Solutions with u(0) = cos(3x)');

subplot(1,2,2)

plot(t,y2), grid on

xlabel('Time (s)');

ylabel('{u(x,t)}');

title('Solutions with u(0) = cos(7x)');

function du = spectral_ivp(t,u)

    global v v0

    Nx = length(u);

    k = (0:Nx-1)';

    du = real((1/Nx)*ifft(-v0*(1j*k).*fft(u) + v*(1j*k).^2 .*fft(u)));

% du = real(-v0*ifft((1j*k)/Nx .*fft(u)) + v*ifft((1j*k).^2 /Nx .*fft(u)));

end

function du = general_ivp(t,u)

    global v v0

    Nx = length(u);

    k = (0:Nx-1)';

    du = (1j*k).*u.*(-v0+v*(1j*k));

end

clc, clear all, close all

%% part 1)

v = 0.1;

Nx = 64;

v0 = 0.5;

tend = 2;

x = linspace(0, 2*pi, Nx);

%% First, solve with u(0) = cos(3x)

u0 = cos(3*x);

[t,y] = PseudoSpectral(Nx, tend, v0, v, u0);

%% Part 2)

[t,y] = CentredFiniteDifference(Nx, tend, v0, v, u0);

plot(t,y)

%% Part 3

Nx = 64;

Nt = 4*Nx;

tend = 2;

x = linspace(0, 2*pi, Nx);

t = linspace(0, tend, Nt);

% Create initial condition

% u0 = cos(3*x);

u0 = exp(-(x-pi).^2 /0.5^2);

% First with v = 1.0

v = 1.0;

[t11,y11] = PseudoSpectral(Nx, tend, v0, v, u0);

[t12,y12] = CentredFiniteDifference(Nx, tend, v0, v, u0);

% Now with v = 0.005;

v = 0.005;

[t21,y21] = PseudoSpectral(Nx, tend, v0, v, u0);

[t22,y22] = CentredFiniteDifference(Nx, tend, v0, v, u0);

% Now using Explicit Euler

% Now plot

figure

subplot(2,2,1)

plot(t11, y11)

xlabel('Time (s)')

title('Pseudo-Spectral Method, v = 1.0');

grid on

subplot(2,2,2)

plot(t21, y21)

xlabel('Time (s)')

title('Pseudo-Spectral Method, v = 0.005');

grid on

subplot(2,2,3)

plot(t12, y12)

xlabel('Time (s)')

title('Finite Difference Method, v = 1.0');

grid on

subplot(2,2,4)

plot(t22, y22)

xlabel('Time (s)')

title('Finite Difference Method, v = 0.005');

grid on