MIT Numerical Methods for PDE Lecture 7: Burgers equation demo

[arjunkc]

Its difficult to find High level problems. Thank you for this

[lilissetyawati]

I'm sorry to distrub you Sir, can i try your script ? I made script by myself but the script is doesnt going well

[AeroCFD]

burgers.m
```
close all;

dt = 0.01;
nt = 200;
nx = 1024;

u = zeros(nt+1, nx) * nan;
subplot(1, 2, 1);
[x, u(1, :)] = drawFunction(0, 3, nx, 'u0');
clf;
for i = 1:nt
u(i+1, :) = u1(end, :);
subplot(1, 2, 1);
plot(x, u(i+1, :), 'r', 'LineWidth', 3);
title(sprintf('t = %f', i * dt));
grid on;
ylim([-1, 1]);
xlabel('x'); ylabel('u');
subplot(1, 2, 2);
contourf(x, dt * [0:nt], u, 30);
colorbar;
axis tight;
xlabel('x'); ylabel('t');
refresh;
pause(0.01);
end
```

drawFunction.m
```
function [x, u] = drawFunction(x0, x1, N, uName, ylimits)
% [x, u] = drawPeriodicFunction(N, uName);
% User draw a periodic function on screen using mouse.
% Inputs N: number of points in the returned function
% uName: name of function (displayed)
% Returns x: N equally spaced points in [0, 2*pi)
% u: function values at x.
xlim([x0, x1]);
if nargin > 4
ylim(ylimits);
else
ylim([-1, 1]);
end
xlabel('x');
ylabel(uName);
grid;

% callback functions are on the bottom of this file
set(gca, 'UserData', [0]); % 0: blank, 1: drawing, 2: done

% wait until mouse up
finished = 0;
while ~finished
data = get(gca, 'UserData');
if length(data) == 0 % user closed figure
x=[]; u=[]; close; return;
end
% check whether "up" has been called
if data(1) == 2
finished = 1;
else
pause(0.1);
end
end

% line coordinates from gca.UserData
[x, u] = interpLinspace(data(2:2:end), data(3:2:end), x0, x1, N);

% replot periodic line
cla;
h = line(x, u);
set(h, 'LineWidth', 3);
set(h, 'color', 'r');

function [x, y] = interpLinspace(x0, y0, xL, xR, N)
if x0(2) - x0(1) > 0 % increasing x
xp = [xL, x0, xR];
else % decreasing x
xp = [xR, x0, xL];
end
yp = [y0(1), y0, y0(end)];

x = linspace(xL, xR, N);
y = interp1(xp, yp, x, 'linear');


function motion(src, data)
data = get(gca, 'UserData');
if length(data) == 0 return; end
if data(1) == 1
pt = get(gca, 'CurrentPoint');
x = pt(1, 1);
y = pt(1, 2);
if x <= 0 || x >= 2*pi return; end
if length(data) < 5 % fewer than 2 points are stored (line segment cannot be plotted)
data = [data, x, y];
else
if data(end-1) == data(end-3) % repeated x coordinates
data = data(1:end-2);
else
% draw line segment
h = line([data(end-3), data(end-1)], [data(end-2), data(end)]);
set(h, 'LineWidth', 3);
set(h, 'color', 'r');
% note that the last line segment is not plotted but this
% effect is negligible

% add to data
if (x - data(end-1))*(data(end-1) - data(end-3)) > 0
data = [data, x, y];
end
end
end
set(gca, 'UserData', data);
end
return

function down(src, data)
data = get(gca, 'UserData');
if length(data) == 0 return; end
data(1) = 1;
set(gca, 'UserData', data);
motion;
return

function up(src, data)
data = get(gca, 'UserData');
if length(data) == 0 return; end
data(1) = 2;
set(gca, 'UserData', data);
motion;
return
```

burgersDdt.m
```
function dudt = burgersDdt( t, u )
n = length(u);
dx = 3 / n;
fc = 0.5 * u.^2;
fb = 0.5 * (fc(1:end-1) + fc(2:end));
a = max(abs(u(1:end-1)), abs(u(2:end)));
d = u(1:end-1) - u(2:end);
L = 2 * d(1:end-2) .* abs(d(3:end)) ./ (abs(d(1:end-2)) + abs(d(3:end)) + 1E-8);
L(d(1:end-2) .* d(3:end) <= 0) = 0;
d(2:end-1) = d(2:end-1) - L;
fd = 0.5 * a .* d;
fb = [0; fb + fd; 0];
if u(1) < 0 fb(1) = fc(1); end
if u(end) > 0 fb(end) = fc(end); end
dudt = (fb(1:end-1) - fb(2:end)) / dx;
end
```