Local vs global truncation error

The truncation error does not satisfy that equation, it's just its definition.

Consider two following problems:

• The first is an ODE. $$ y'(t) = f(t, y(t))\\ y(0) = a. $$ Its solution is some smooth function $y(t)$.
• The second is a difference equation $$ \frac{z_{i+1} - z_i}{h} = f(t_i, z_i)\\ z_0 = a. $$ Its solution is some discrete function $z_i$.

I've intentionally used different letters to denote those two solutions. They are quite different, the former is a smooth function while the latter is a discrete one. One needs to be careful even to compare those two. Usually the third function is introduced. It is defined as a restriction of the smooth $y(t)$ to the grid $t_i$, where the discrete function $z_i$ is defined. Let's denote the restriction as $w_i$: $$ w_i \equiv y(t_i). $$

The function $w_i$ is discrete just like $z_i$ and $w_i$ coincide with $y(t)$ at grid points. Since now $w_i$ and $z_i$ are functions of the same class we can easily compare them: $$ e_i = w_i - z_i \equiv y(t_i) - z_i. $$ So, roughly speaking, the global error shows how close are $y(t)$ and $z_i$ (by restricting the former to the grid). When someone is solving some problem numerically the global error is what he is interesting in. Anyway, direct computation of global error is almost impossible, since we often simply do not have the exact values of $w_i = y(t_i)$ ( in contradistinction to $z_i$, which we can compute easily).

And the local truncation error concept comes to the rescue. Note that previously we've compared the solutions. Now we're going to compare problems. Take $z_i$. It is the solution to the second problem. Plugging $z_i$ into it makes it a valid identity $$ \frac{z_{i+1} - z_i}{h} = f(t_i, z_i)\\ z_0 = a. $$ But if we now take $w_i$ and try to plug it into the difference scheme we wont get an identity. Instead we'll get a residual: $$ \frac{w_{i+1} - w_i}{h} = f(t_i, w_i) \color{red}{{}+ d_i}\\ w_0 = a \color{red}{{} + d_0}. $$ If we are very lucky, some residuals may vanish, like $d_0$, but often it is not the case.

So why is $d_i$ interesting while it also is defined in terms of $w_i$ (the unknown solution to the original problem)? It turns out that we can estimate the $d_i$ without knowing the exact values of $w_i$ by just knowing the original problem. $$ d_i = \frac{w_{i+1} - w_i}{h} - f(t_i, w_i) \equiv \frac{y(t_{i+1}) - y(t_i)}{h} - f(t_i, y(t_i)) = \\ = y'(t_i) + h \frac{y''(t_i)}{2} + O(h^2) - f(t_i, y(t_i)) = \\ = \color{blue}{\left[y'(t_i) - f(t_i, y(t_i))\right]} + \color{red}{h \frac{y''(t_i)}{2} + O(h^2)} $$ The blue term in braces is exactly the original ODE, and $y(t)$ is exactly its solution. So the term is equal to zero. $$ d_i = h \frac{y''(t_i)}{2} + O(h^2). $$ Similar result may be obtained if using different form of Taylor's formula: $$ d_i = h \frac{y''(\xi_i)}{2}, \qquad \xi_i \in [t_{i}, t_{i+1}]. $$

So now we can estimate the local truncation error, but we're interested in the global error.

To relate them we need to introduce another concept of stability. Consider the two discrete problems $$ \begin{aligned} &\frac{z_{i+1} - z_i}{h} = f(t_i, z_i)\\ &z_0 = a \end{aligned} \qquad\text{and}\qquad \begin{aligned} &\frac{w_{i+1} - w_i}{h} = f(t_i, w_i) \color{green}{{} + d_i}\\ &w_0 = a \color{green}{{} + d_0} \end{aligned}. $$ Pretend that we know $d_i$. Let's view the second problem as a perturbation of the first one. That's reasonable, since $d_i$ is a small value of $O(h)$ magnitude. A difference problem is called stable if such small perturbations result in small changes of the solution. For this case this means that the difference $z_i - w_i$ will also be small. Precisely $$ \max_i |z_i - w_i| \leq C \max_i |d_i| $$ where $C$ is called the stability constant of the method. For the explicit Euler method it can be shown that for Lipschitz-continuous $f$ $$ C \leq e^{LT} $$ with $L$ being the Lipschitz constant of $f$ and $T$ is the total integration time $T = \max_i t_i$.

Finally we can relate the global error and the local truncation error by $$ |e_i| \leq C \max_i |d_i| $$

If the local truncation error tends to zero when the discrete mesh is refined the numerical method is called consistent. The Lax theorem states that a stable consistent method converges, in sense that $e_i \to 0$ when the mesh is refined.

Another nice picture to connect the local and global errors is to take the points $(t_n,y_n)$, $n=0,1,...,N$ of a Runge-Kutta method and to consider the family of exact solutions to the problems $$ (y^n)'=f(t,y^n) \text{ with initial values } y^n(t_n)=y_n. $$

By the assumption of a global Lipschitz constant and via the Gronwall lemma, we know that the development of the distance of the solutions in this family is bounded by an exponential with factor $L$, for instance $$ \|y^{n+1}(t_N)-y^n(t_N)\|\le e^{L(t_N-t_k)}\|y^{n+1}(t_k)-y^n(t_k)\| $$ Using $k=n$ for the explicit or $k=n+1$ for the implicit Euler method allows to employ relatively simple formulas for the local error.

Given that the local error terms are bounded in terms of local truncation errors by $|t_{n+1}-t_n|\max_j|d_j|$ one can assemble these propagated local error terms into the global truncation error as in \begin{align} \|y_{N}-y(t_N)\|&=\|y^N(t_N)-y^0(t_N)\| \\ &\le\sum_{n=0}^{N-1}\|y^{n+1}(t_N)-y^n(t_N)\| \\ &\le\sum_{n=0}^{N-1}e^{L(t_N-t_{n+1})}\|y_{n+1}-y^n(t_{n+1})\| \\ &\le\sum_{n=0}^{N-1}e^{L(t_N-t_{n+1})}|t_{n+1}-t_n|\max_j|d_j| \\ &\le\int_{t_0}^{t_N}e^{L(t_N-s)}ds\max_j|d_j| \\ &=\frac1L(e^{L(t_N-t_0)}-1)\max_j|d_j| \end{align} which gives a bound for the global truncation error composed from the local truncation errors.

For a $p$th order method, and equidistant time points $t_k=t_0+kh$, the local errors $y_{n+1}-y^n(t_{n+1})$ resp. $\Bigl(y(t)+h\Phi_f(t,y(t),h)\Bigr)-y(t+h)$ are of order $O(h^{p+1})$ so that for the local truncation error one gets $d_k=O(h^p)$ which consequently is also the global error order.

On the other hand the general bound above is strictest if the local truncation errors are all almost equal which then gives the general strategy for step size adaptation. Estimate the constant in $d_k=c_k·(t_{k+1}-t_k)^p$ and compute $t_{k+1}$ as $t_k+\sqrt[p]{\varepsilon_{\rm local}/c_k}$, where the local error $ε_{\rm local}$ is estimated as $ε_{\rm global}/(t_N-t_0)$ or, if an estimate of the Lipschitz constant is known, as $ε_{\rm global}·L/(\exp(L(t_N-t_0))-1)$.

Two quite exhaustive answers have already been given.

Local error is the error you get from your truncated approximation of the function of interest and is thus the error you get in a single time step from using your propagator function.

Global error is the accrued error you get from propagating your truncated solution many times and is thus defined for multiple time steps.

I really just want to add a reference that helped me understand it and a nice derivation of it for the Euler method case.By what I understand the procedure depicted in the document is also the general procedure for deriving the global error for a specific method, you try and relate $e_{k+1}$ to $e_{k}$ and in this way derive a bound. This is also a nice video.

P.S. This my first answer on stack and I am also quite new to numerics, please edit or remove if inappropriate.