I am currently reading Numerical Partial Differential Equations: Finite Difference Methods by J.W. Thomas and I'm trying to get a better grasp of the following definition on page 42.
Definition 2.2.1. A difference scheme $L_k^n u_k^n = G_k^n$ approximating the partial differential equation $\mathcal{L} v = F$ is called pointwise convergent if for any $x$ and $t$, as $(k \Delta x, (n+1)\Delta t)$ converges to $(x,t)$, $u_k^n$ converges to $v(x,t)$ as $\Delta x$ and $\Delta t$ converge to $0$.
In this context $v:[0,L] \times [0,T] \to \mathbb{R}$ denotes the true solution and $u_k^n$ denotes the numerical approximation of $v(k \Delta x, n \Delta t)$.
My questions.
How am I to interpret "as $(k\Delta x, (n+1)\Delta t)$ converges to $(x,t)$" ? For a fixed $(x,t)$, the values of $k$ and $n$ must depend on $\Delta x$ and $\Delta t$, so I find this sentence confusing. My attempts to paraphrase it are below, but I am not sure if I'm interpreting things correctly.
In a remark on page 44, the author makes the following remark comparing the above definition with a definition of convergence in terms of norms (definition 2.2.2):
It must be made clear how Definition 2.2.2 differs from Definition 2.2.1. Using Definition 2.2.1, the rate at which $u_k^n$ converges to $v(x,t)$ (rate in terms of $\Delta x \to 0$ and $\Delta t \to 0$) can vary greatly for different values of $x$. In fact, Definition 2.2.1 would allow for a scheme to converge for some values of $(x,t)$ and not others...
The last sentence does not make sense to me. How could a pointwise convergent numerical scheme not converge for some values of $(x,t)$? Perhaps the author made a typo, and was actually referring to Definition 2.2.2?
Here are my attempts to interpret and paraphrase Definition 2.2.1:
Definition 1. The scheme $L_k^n u_k^n = G_k^n$ is pointwise convergent at $(x,t) \in [0,L] \times [0,T]$ if, for any sequence of discretization parameters $((\Delta x_m, \Delta t_m))_{m=1}^{\infty}$ converging to $(0,0)$, and any sequence of lattice points $((k_m, n_m))_{m=1}^{\infty}$ satisfying
- $0 \leq k_m \leq \lfloor L/\Delta x_m \rfloor \; \text{ and } 0 \leq n_m \leq \lfloor T/\Delta t_m \rfloor$ for all $m$, and
- $\lim\limits_{m \to \infty} (k_m \Delta x_m, n_m \Delta t_m) = (x,t)$,
we have $$\lim\limits_{m \to \infty} u(k_m \Delta x_m, n_m \Delta t_m) = v(x,t).$$
The scheme is pointwise convergent if it is pointwise convergent at every $(x,t)$ in $D$.
Definition 2. A difference scheme $L_k^n u_k^n = G_k^n$ approximating the partial differential equation $\mathcal{L} v = F$ is a pointwise convergent scheme if for each pair of discretization parameters $\Delta x, \Delta t$ and each pair $(k,n)$ such that $0 \leq k \leq \lfloor L/\Delta x \rfloor$ and $0 \leq n \leq \lfloor T/\Delta t \rfloor$, there exists a constant $C_{k,n}$ independent of $\Delta x, \Delta t$ such that
$$ |u_k^n - v(k \Delta x, n \Delta t)| \leq C_{k,n} (\Delta x + \Delta t). $$
And if the last definition is correct (or almost correct), I'm not sure whether the constant $C_{k,n}$ should be allowed to depend on $k$ and $n$, or if it should be independent of all parameters. If the latter, then the definition would seem to be a type of uniform convergence, which is stronger than Definition 2.2.1...