https://github.com/mathworks/simple-heat-equation-solver

Simple Heat Equation solver using finite difference method
https://github.com/mathworks/simple-heat-equation-solver

Last synced: about 2 months ago
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Simple Heat Equation solver using finite difference method

• Host: GitHub
• URL: https://github.com/mathworks/simple-heat-equation-solver
• Owner: mathworks
• License: other
• Created: 2019-12-06T06:32:11.000Z (over 5 years ago)
• Default Branch: master
• Last Pushed: 2023-03-18T06:02:41.000Z (over 2 years ago)
• Last Synced: 2025-05-06T00:56:07.069Z (about 2 months ago)
• Language: MATLAB
• Size: 5.44 MB
• Stars: 20
• Watchers: 3
• Forks: 11
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: license.txt
  • Security: SECURITY.md

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README

        
# Finite differences for the 2D heat equation

[![Open in MATLAB Online](https://www.mathworks.com/images/responsive/global/open-in-matlab-online.svg)](https://matlab.mathworks.com/open/github/v1?repo=mathworks/Simple-Heat-Equation-solver)

[![View Simple Heat Equation solver on File Exchange](https://www.mathworks.com/matlabcentral/images/matlab-file-exchange.svg)](https://jp.mathworks.com/matlabcentral/fileexchange/59916-simple-heat-equation-solver)

Implementation of a simple numerical schemes for the heat equation.



Applying the second-order centered differences to approximate the spatial derivatives,



Neumann boundary condition is employed for no-heat flux, thus please note that the grid location is staggered. Once the right hand side is obtained, the equation can be solved by the ODE suite. Here we use ode15s. Copyright 2015-2016 The MathWorks, Inc.

![image_0.png](SimpleHeatEquation_images/image_0.png)

# Problem Setup

```matlab

N = 50; % Number of grid in x,y-direction

L = 4*pi; % Domain size

% Grid point

x = linspace(0,L,N);

y = linspace(0,L,N);

% Make it staggered.

x = (x(1:end-1)+x(2:end))/2;

y = (y(1:end-1)+y(2:end))/2;

[X,Y] = meshgrid(x,y);

```

# Initial Condition

```matlab

% Let's use MATLAB logo.

% A variable u0 is defined at the center of each grid cell

% thus the number of grid point is N-1.

u0(:,:) = peaks(N-1);

% Plot it

handle_surf = surf(X,Y,u0);

handle_axes = gca;

handle_axes.ZLim = [-10,10];

handle_axes.CLim = [-10,10];

title('Evolution of MATLAB Logo by Heat equation');

```

![figure_0.png](SimpleHeatEquation_images/figure_0.png)

```matlab

```

# Simulation

```matlab

dx = x(2)-x(1); % spatial grid size

alpha = 2; % coefficient

tspan = linspace(0,1,40);

[t,u] = ode15s(@(t,x)getRHS(x,alpha,dx,N),tspan,u0(:));

```

# Visualize

```matlab

Tn = length(t);

u = reshape(u,Tn,N-1,N-1);

filename = 'heat.gif';

for ii=1:Tn

    Z = u(ii,:,:);

    Z = squeeze(Z);

    handle_surf.ZData = Z;

    drawnow;

    frame = getframe(gcf);

    im = frame2im(frame);

    [A,map] = rgb2ind(im,256);

    if ii==1

        imwrite(A,map,filename,'gif','LoopCount',Inf,'DelayTime',0.05);

    else

        imwrite(A,map,filename,'gif','WriteMode','append','DelayTime',0.05);

    end

end

```

![figure_1.png](SimpleHeatEquation_images/figure_1.png)

Copyright 2015-2016 The MathWorks, Inc.

[![View Simple Heat Equation solver on File Exchange](https://www.mathworks.com/matlabcentral/images/matlab-file-exchange.svg)](https://jp.mathworks.com/matlabcentral/fileexchange/59916-simple-heat-equation-solver)