Eigenvalues as a function of k for the linear diffusion equation with various time stepping schemes. Dimensionless timestep n = 2*dt*K/dx^2, where K is the diffusion constant and dx is the grid spacing. In these units, the maximum stable FTCS timestep is:
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To plot wavenumber on the x-axis, I define
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Forward time, centered space (FTCS):
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Midpoint method with perturbed halfstep, dt*(1+d)/2:
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![y=1+n*[cos(k)-1+[1+d]*n/4*[cos(2*k)-(4*cos(k))+3]]](formula9.png)
Implicit time, centered space (ITCS):
![y=1/(1+n*[1-cos(k)])](formula11.png)
Implicit midpoint method. This time, the intermediate timestep is (1+c)/2:
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![y=1/(1-(n*[cos(k)-1-([1+c]*n/4*[cos(2*k)-(4*cos(k))+3])]))](formula14.png)
Analytic result:

Nyquist Frequency:
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