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3

It's hard to tell from your phrasing and because your link is broken, but are your boundary conditions periodic or not? If the problem is periodic, then spectral methods are the way to go since Fourier series (and their discrete coutnerparts) converge much faster for periodic functions than for more general functions. For a $1$-D problem like this, a ...

1

So I've done a modified version of this equation for a different goal, and we usually use a randomized starting profile to show that it converges to an approximate steady state regardless of the initial conditions. It really depends on the type of situation you want to simulate. I like the randomization of the initial conditions, because I think that for ...

2

The Newton-Raphson method can be used to solve non-linear systems of equations. The first step is to write your system as a root finding problem: $$f(T_n) = \left( \frac{C}{\Delta t} + K \right) T_n - Q_n - \frac{C}{\Delta t} T_{n-1} = 0$$ Taylor expand this equation about an initial guess $T^0_n$, keeping only the linear term:  f(T_n) \approx f(T^0_n)...

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