Consistency Versus Convergence, seeking intuition

[fahraynk]

TL;DR

Trying to understand the difference between consistency and convergence in PDE finite difference methods, and why we need both definitions. Why they do not really both mean the same thing.

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 Δ𝑡

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+Δ𝑡)
with the finite difference scheme and having a low error, while consistency is plugging the true values of 𝑢 and 𝑢(𝑡+Δ𝑡)
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+Δ𝑡)
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?

 

[Periwinkle]

I think convergence means that using the finite difference scheme your solution will converge to a real solution. That's what you thought.

However, consistency does not apply to partial differential equations, but to partial differential operators. The finite difference scheme is consistent with the partial differential equation if, for any smooth function, the value calculated by the finite difference scheme converges to the real value calculated by the operator. Look for convergence on the following page.

 

[fahraynk]

Thank you so much for the reply,

I think convergence means that using the finite difference scheme your solution will converge to a real solution. That's what you thought.

However, consistency does not apply to partial differential equations, but to partial differential operators. The finite difference scheme is consistent with the partial differential equation if, for any smooth function, the value calculated by the finite difference scheme converges to the real value calculated by the operator. Look for convergence on the following page.

Whats the difference between a partial differential operator and a PDE?
Isn't an operator, say, $L$, such that $Lu=u_{tt}+u_{xx}+u_{xt}+...$ $(meaning Lu$ equals the pde?)

The operator produces a true solution right? So $Lu=u^{n+1}$ where $n$ is time. So, basically saying that $Lu$ converges to the finite difference scheme (call it $FD(u)$) means that $|(L-FD)u|\rightarrow 0$ right? BUT $Lu$ is the true solution and $FD(u)$ is the finite difference solution. So.. why is this different than convergence? $|L(u)-FD(u)|\rightarrow 0$ is convergence right?