Local truncation error

Local truncation error is the error a numerical method makes in one step when it uses the exact previous value. In Linear Algebra and Differential Equations, it shows how well a Runge-Kutta or multistep method matches the true solution.

What is local truncation error?

Local truncation error is the error made in a single step of a numerical method for a differential equation, assuming the starting value for that step is exact. In other words, it asks, “If I begin this step from the true solution, how far off is the method after one update?” That makes it a step-by-step accuracy measure, not the total error after many steps.

This comes up when you approximate an initial value problem like y' = f(t, y) with a formula that moves from t_n to t_{n+1}. For a method such as Euler’s method, Runge-Kutta, or a multistep method, you do not solve the equation exactly each time. You use a rule based on slopes, weighted slopes, or past values. Local truncation error measures the mismatch created by that one rule.

A useful way to think about it is this: local truncation error isolates the method itself from the errors that build up over time. If you plug the exact solution into the numerical formula, the leftover difference is the local truncation error. That is why Taylor series are often used to analyze it. Expanding the exact solution and the numerical method side by side shows which terms match and which terms are missing.

The size of the local truncation error usually depends on the step size h. Smaller h usually means a smaller one-step error, but the exact rate depends on the method’s order. For example, a higher-order Runge-Kutta method matches more Taylor terms, so its local truncation error shrinks faster as h gets smaller.

Do not mix this up with global error. Local truncation error is about one step in isolation. Global error is what you actually have after many steps, when those one-step errors plus any propagation through the method all pile up. In multistep methods, that distinction matters even more because an early small error can affect later values.

Why local truncation error matters in Linear Algebra and Differential Equations

Local truncation error tells you how good the update rule is before errors start piling up across many steps. In Differential Equations, that gives you a cleaner way to compare methods like Euler, Runge-Kutta, and linear multistep formulas. If two methods use the same step size, the one with smaller local truncation error is usually matching the true curve more closely at each move.

It also connects directly to order of accuracy and order of convergence. When you see a method described as first-order, second-order, or fourth-order, that label is tied to how fast its truncation error goes to zero as h shrinks. So local truncation error is not just a leftover term from a Taylor expansion, it is the feature that tells you whether the method is crude, decent, or very precise for a given step size.

In a Linear Algebra and Differential Equations course, this shows up when you analyze numerical solutions for initial value problems, compare formulas in class, or justify why a method needs a smaller step size to stay reliable. It also helps you explain why Runge-Kutta methods often give better results than simpler single-step methods, even if they cost more arithmetic per step.

How local truncation error connects across the course

global truncation error

Local truncation error measures the error from one idealized step, while global truncation error is the total error after many steps. They are connected, but they are not the same thing. A method can have a small local error and still drift if those step-by-step errors accumulate or if the method amplifies them.

order of convergence

Order of convergence describes how fast a numerical method approaches the true solution as the step size shrinks. Local truncation error helps determine that rate. If you can show the one-step error is proportional to a power of h, that power is part of the story behind the method’s convergence behavior.

stability

Stability asks whether small errors stay controlled as you keep stepping forward. Local truncation error measures the size of the error introduced at each step, but stability tells you what happens to those errors afterward. A stable method can keep small local errors from exploding into a bad approximation.

error estimate

An error estimate is the practical number you use to judge how far your numerical answer might be from the true one. Local truncation error is one piece of that picture because it gives a built-in estimate of the method’s accuracy per step. In homework, it often appears when you compare formulas or choose a step size.

Is local truncation error on the Linear Algebra and Differential Equations exam?

A problem set question might ask you to find the local truncation error of a method by substituting the exact solution into the update formula and expanding with Taylor series. You may also be asked to compare two methods and identify which one has higher order, smaller one-step error, or better accuracy for the same step size. In multistep problems, watch for the phrase “assuming previous values are exact,” because that is the clue that the question is about local truncation error, not global error. If you are checking work, the common mistake is to use the computed approximate values instead of the exact solution when evaluating the one-step error. That changes the problem entirely. On quizzes and homework, you may also explain why a smaller step size improves the approximation and connect that to the order of the method.

Local truncation error vs global truncation error

These are easy to mix up. Local truncation error is the error created in one step, assuming everything before it is exact. Global truncation error is the accumulated difference between the numerical solution and the true solution after many steps, so it includes the effect of repeated local errors.

Key things to remember about local truncation error

• Local truncation error is the one-step error made by a numerical method when the previous value is assumed to be exact.

• It is found by comparing the exact solution to the numerical update formula, often using Taylor series.

• Smaller step sizes usually reduce local truncation error, but the exact rate depends on the method’s order.

• Local truncation error is different from global error, which measures the total error after many steps.

• In Runge-Kutta and multistep methods, local truncation error helps you compare accuracy and choose a step size.

Frequently asked questions about local truncation error

What is local truncation error in Linear Algebra and Differential Equations?

It is the error a numerical method makes in one step when the starting value for that step is exact. You use it to judge how well a method like Euler, Runge-Kutta, or a multistep formula follows the true solution of a differential equation. It is a step-level measure, not the total accumulated error.

How do you find local truncation error?

You plug the exact solution into the numerical method and see what leftover term remains after one step. Taylor series are usually the easiest way to do this because they let you compare the exact solution and the method term by term. The first term that does not match tells you the size and order of the error.

Is local truncation error the same as global error?

No. Local truncation error is the error from a single step, while global error is the overall difference after many steps. Global error includes the local errors plus any way the method carries those errors forward, which is why it is usually larger.

Why does step size affect local truncation error?

The step size h controls how far the method jumps each time. When h gets smaller, the Taylor series terms that drive the error usually get smaller too, so the one-step approximation improves. That is why a method often becomes more accurate when you use more, smaller steps.