Recently, I get a question about the convergence of some implicit difference scheme (e.g., Crank-Nicolson method) for (1D) time-dependent PDEs, such as heat equation and convection-diffusion equation. More precisely,
Suppose we use the CN scheme for 1d diffusion equation at time interval [0,T], (generally) we also use the \delta t = \tau = T/M, M is the number of time steps. t_{j+1} = t_j + \tau, i.e., 0 = t_0 < t_1 < ....< t_M = T. then it is not easy for us to prove the convergence and stability of the CN scheme. However, if we use the variable time step size \tau_j (j = 0,1,...,M-1), is it still easy for us to prove the convergence of the CN scheme with the variable time step size? or how to prove this convergence of this CN scheme (via matrix method)?
@Kaveh Zamani Thanks for your answer, I got it. For me, recently I consider the 1d PDEs (like the convection-diffusion equations), then I wonder what happens if we use the variable time step size. In most cases, we like to use the equivalent time step size.
@Kaveh Zamani, "equivalent time step" just means that we choose \tau = T/N_t, i.e., \tau is the time stepsize. Maybe, can I send you a short file for describing my ideas?
For heat conduction Fourier's equation applies. As it is linear and has constant coefficients, the analyses you wish to do are well-known in many text books. Indeed, as long as it is Fourier's equation with a constant diffusivity then you can find the convergence and stability properties also in 2D and 3D, for many different methods: forward difference (Euler), backward Euler, Crank-Nicholson, even more exotic types such as the PDE version of BDF2 [backward Euler is formally BDF1].. Much of this will be found in the appropriate numerical analysis books for PDEs. I haven't looked, but I guess that a simple google search might also uncover someone's lecture notes - I have found many others online from many parts of the world!
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