Calculus Without Tears

Computation

The FDM substitutions

The solution for a time dependent 2-D problem is computed for a set of grid points that cover the problem domain, a representative grid is shown in the diagram


The model for time dependent 2-D heat transfer is

Note that temperature in the plate is a function of three variables, T(t, x, y), and taking samples at discrete times and grid points gives values for T(n, i, j). The standard notation for the sampled values is Tⁿ(i,j).

We make the usual Euler's substitution for the time derivative


The FDM substitutions for the spatial derivatives are


Making both substitutions into the differential equation model gives


and we can rearrage the equation to solve for Tⁿ⁺¹(i,j)

and we're good to go.

The code

With dx = dy and C = dt*k / (c_p*p*dx²), the code is

for n = 1:N
   Tn = T; % save Tn before calculating Tn+1
   for i = 2:sizeX - 1 % note that boundary points are omitted
      for j = 2:sizeY - 1
         T(i,j) = T(i,j) + C*(Tn(i+1,j) + Tn(i-1,j) + Tn(i,j+1) + Tn(i,j-1) - 4*T(i,j));
      end
   end
   % apply boundary conditions here
end

Simple as that !

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