Global error

Global error is the accumulated difference between the exact solution of a differential equation and the numerical solution after many steps. In Linear Algebra and Differential Equations, it measures how far your approximation drifts over the whole interval.

What is global error?

Global error is the total error in a numerical solution of a differential equation after you have marched across the whole interval. If you solve an IVP with step size h, global error is the gap between the exact value y(t_n) and the computed value y_n at a later step.

In this course, that means you are not looking at just one bad step. You are looking at the effect of every step that came before it. A small local mistake can get carried forward, reshaped, or sometimes damped out by the method and the differential equation itself.

That is why global error is tied to both accuracy and stability. A method can have a small local truncation error and still produce a poor answer if the method is unstable for the problem you are solving. On the other hand, a stable method can keep the accumulated error under control as long as the step size is reasonable.

This is especially noticeable in multistep methods, where each new approximation uses several previous values. That reuse makes the method efficient, but it also means past errors get recycled. If the method is high order, the global error usually shrinks faster as h gets smaller. If the method is unstable, making h smaller does not automatically fix the result.

A compact way to think about it is this: local error is what one step does wrong, while global error is what the whole numerical process leaves you with at the end. For example, if Euler's method keeps underestimating the slope on each step, those underestimates pile up, and the final answer can drift far from the true solution even when each individual step looks only a little off.

In practice, you check global error by comparing numerical output at several step sizes, watching whether the approximations settle toward a consistent answer. If they do not, the issue may be instability, not just step size.

Why global error matters in Linear Algebra and Differential Equations

Global error is what tells you whether a numerical method is giving you a trustworthy answer for a differential equation over time. In Linear Algebra and Differential Equations, you rarely solve every IVP exactly, so you need to know whether the approximation stays close to the true solution or slowly drifts away.

This matters in multistep methods because the whole point of the method is to reuse earlier values. That saves work, but it also means the method can carry old mistakes forward. If you are solving a model over many time steps, a tiny mismatch at the start can become a noticeable shift later.

Global error also connects directly to stability analysis. A method with a good order of accuracy can still behave badly on a stiff equation or on a problem where the step size is too large. So when you see a numerical solution that suddenly blows up, wiggles, or moves away from the expected behavior, global error is one of the first things to suspect.

You will also see this idea when comparing methods. If one method has order 1 and another has order 2, the higher-order method should usually reduce global error faster as you shrink h. That gives you a practical way to judge whether a method is worth using for a homework problem, a computational lab, or a long-run model.

How global error connects across the course

local error

Local error is the mistake made in one step of a numerical method, while global error is the accumulated effect after many steps. If a method has small local error but the solution still drifts, you are seeing how those small step-by-step mistakes build up across the interval.

stability

Stability tells you whether a numerical method keeps errors under control as you continue stepping forward. A method can have a good formula on paper and still produce a bad approximation if errors grow instead of shrinking, which makes global error much larger than expected.

Order of Accuracy

Order of Accuracy describes how fast the error decreases when you make the step size smaller. In many differential equation methods, higher order means global error drops more quickly as h decreases, so you can get a much better approximation without taking an extreme number of steps.

convergence

Convergence is the idea that a numerical solution gets closer to the exact solution as the step size shrinks. Global error is one of the main things you watch to see whether that is actually happening in a method, especially over long intervals.

Is global error on the Linear Algebra and Differential Equations exam?

A problem set question may give you a numerical method and ask whether the approximation is staying close to the true solution as h changes. You might compare two step sizes, estimate the error at a later point, or explain why a method looks accurate for the first few steps but becomes unreliable later. The move is usually to connect the size of the global error to stability and order of accuracy, not just to one isolated calculation. If a method is multistep, be ready to explain how earlier rounding or truncation errors can carry into later values.

Global error vs local error

Local error is the error from one step of a numerical method. Global error is the total error after all the steps from the starting value to the point you are checking. A common mistake is treating them as the same thing, but a method can have small local error and still end with a noticeable global error.

Key things to remember about global error

• Global error is the accumulated difference between the exact solution and the numerical approximation after many steps.

• In differential equations, it is what you watch when deciding whether a method stays reliable across an entire interval.

• Small local errors can add up, and stability determines whether a method controls that buildup or lets it grow.

• Higher order methods usually reduce global error faster as the step size gets smaller, but only if the method is stable for the problem.

• If your approximation starts drifting or blowing up, the issue may be global error rather than a single bad arithmetic step.

Frequently asked questions about global error

What is global error in Linear Algebra and Differential Equations?

Global error is the difference between the exact solution of a differential equation and the computed numerical solution after you have taken many steps. It measures the total drift across the whole interval, not just the mistake from one step. That makes it a better picture of how accurate your approximation really is.

How is global error different from local error?

Local error comes from one step of a method, while global error is the accumulated result after all the steps. A method can look fine locally and still end up far from the true solution if the errors keep building up. That is why stability matters so much.

Why does step size affect global error?

Smaller step sizes usually make each step more accurate, so the total global error often goes down. But step size is not the whole story, because an unstable method can still behave badly even when h is small. That is why you look at both accuracy and stability together.

How do you know if a numerical method has too much global error?

A common check is to run the method with smaller step sizes and see whether the answers settle toward one value. If the approximations keep wandering, growing, or oscillating strangely, the global error may be too large. In class problems, this often shows up when comparing a method’s output to an exact solution or a more accurate approximation.