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If a spatial grid is given with time levels like this: enter image description here

to solve the following model problem

enter image description here

Now consider the following discretization schemes:

Scheme 1

enter image description here

Scheme 2

enter image description here

Usually, to determine order of accuracy, i.e., $O(\Delta x^2, \Delta t)$ etc, I know that I must expand the discretized quantities in Taylor's series about the point $(c, 1)$ or $(c, 0)$ etc.

Q1) But in this case If I am simply asked about the order of accuracy of the schemes 1 and 2, how do I proceed ? In Scheme 1, at least I have a hint that I must expand around $(c,1)$. But what point should I expand around for Scheme 2?

Q2) Are both schemes "Crank-Nicolson" for B?

Q3) In scheme 2, the approximation for C has 4 terms --is that also "Crank-Nicolson" ?

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  • $\begingroup$ unless you or someone else has proven these methods are a given order of accuracy you can't really tell. You could "cheat" by just implementing and testing the methods. $\endgroup$ Commented Jun 8 at 14:55
  • $\begingroup$ I think you might be referring to order of convergence, but I am talking about simple analysis using Taylor's series. $\endgroup$
    – me10240
    Commented Jun 8 at 17:57
  • $\begingroup$ If you can apply your finite-difference operator (as a "black box") to a known analytic function and analyze the results, that should be enough to infer the approximation order. For example, you apply your first-derivative operator D1 to function sin(x) and see how the difference between the result and the exact answer cos(x) scale with the grid spacing. $\endgroup$ Commented Jun 8 at 18:07
  • $\begingroup$ @me10240 Are you talking about local truncation error or global error? Talking about the second one is much harder as it toghtly relates to the convergence of the scheme (unless one can identify the schemes above as known methdos) $\endgroup$
    – Rigel
    Commented Jun 9 at 6:42

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