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?

  • $\begingroup$ scicomp.stackexchange.com/questions/7780/… can't this help? what is A[T] and I[T] btw? $\endgroup$
    – VorKir
    Dec 9 '16 at 20:30
  • 2
    $\begingroup$ What does EBM stands for? $\endgroup$
    – nicoguaro
    Dec 9 '16 at 20:31
  • 3
    $\begingroup$ 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. $\endgroup$ Dec 9 '16 at 22:20
  • $\begingroup$ @WolfgangBangerth I think you mean "a nonlinear system of algebraic equations". $\endgroup$ Dec 10 '16 at 18:15
  • 1
    $\begingroup$ @DavidKetcheson, well that depends on whether you discretize first in time then space, or the other way around :-) $\endgroup$ Dec 10 '16 at 21:05

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.