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What is the definition of consistency?

I have seen a proof that shows a finite difference scheme is consistent, where they basically plug a true solution $u(t)$ into a finite difference scheme, and they get every term, for example $u^{i+1}_{j}$ and $u^i_{j+1}$, using taylors polynomials.

Then, they show that the taylor approximations plugged into the finite difference scheme go to zero as $\Delta t$ goes to zero.

So this seems like the definition should be, Consistency : The error of the real solution in the finite difference scheme goes to zero as time step goes to zero.

So, on the difference between convergence and consistency, it seems like convergence is computing $u(t+\Delta t)$ with the finite difference scheme and having a low error, while consistency is plugging the true values of $u$ and $u(t+\Delta t)$ into the finite difference scheme and having a low error.

But... they are kind of exactly the same, since if the finite difference scheme is convergent, the approximation for $u(t+\Delta t)$ converges to the true value, and so if you plug the true value into the fin. dif. equation or you plug the approximation in, why would the output be different?

Can anyone give me some intuition?

Frank
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1 Answers1

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Consistency is related to the local truncation error (LTE), which is determined by applying the scheme once while assuming that the PDE is satisfied. The LTE is required to vanish as the discretization is refined. Convergence relates to the global error, i.e., the error made by the numerical method over the whole domain of integration. A method can be consistent with the PDE but unstable (explosion of the numerical solution in finite time). For example, this is the case with the centered explicit finite-difference approximation $$\frac{u_i^{n+1}-u_i^n}{\Delta t} + a\frac{u_{i+1}^{n}-u_{i-1}^n}{2\Delta x} = 0$$ of the advection equation $u_t + a u_x = 0$. Such a method is consistent with the PDE by construction, but it is not convergent.

EditPiAf
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