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I am trying to solve numerically the following 1D EBM:

$C\frac{\partial T[x,t] }{\partial t} - \frac{\partial }{\partial x}\left ( D(1-x^2)\frac{\partial T[x,t] }{\partial x} \right ) + I[T] = S[x,t](1-A[T])$

where $C$ and $D$ are constants.

I want to use the Crank-Nicolson method to solve it but I am unsure how to implement it with all of the non-linear terms.

What would the Crank-Nicolson discretization of this equation be?

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  • $\begingroup$ scicomp.stackexchange.com/questions/7780/… can't this help? what is A[T] and I[T] btw? $\endgroup$
    – VorKir
    Dec 9, 2016 at 20:30
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    $\begingroup$ What does EBM stands for? $\endgroup$
    – nicoguaro ♦
    Dec 9, 2016 at 20:31
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    $\begingroup$ The C-N discretization will simply yield a nonlinear differential equation at each time step. You then need to solve it using something like a Newton method. $\endgroup$
    – Wolfgang Bangerth
    Dec 9, 2016 at 22:20
  • $\begingroup$ @WolfgangBangerth I think you mean "a nonlinear system of algebraic equations". $\endgroup$
    – David Ketcheson
    Dec 10, 2016 at 18:15
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    $\begingroup$ @DavidKetcheson, well that depends on whether you discretize first in time then space, or the other way around :-) $\endgroup$
    – Wolfgang Bangerth
    Dec 10, 2016 at 21:05

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Anyone still interested in an answer, see this article:

P. Y. P. Chen and B. A. Malomed, "Lanczos-Chebyshev pseudospectral methods for wave-propagation problems," Mathematics Computers Simulation, vol. 82, no. 6, pp. 1056–1068, Feb. 2012.

In this published method, Crank-Nicholson forward difference method is used with an inner iterative step to cater for the nonlinear terms.

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