Linear multistep method

Key takeaways

Linear multistep method belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Linear multistep method to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear multistep method from memory before moving on to harder problems.

Reference excerpt

Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of linear multistep methods, a linear combination of the previous points and derivative values is used.

Definitions Numerical methods for ordinary differential equations approximate solutions to initial value problems of the form

y ′ = f ( t , y ) , y ( t 0 ) = y 0 . {\displaystyle y'=f(t,y),\quad y(t_{0})=y_{0}.}

The result is approximations for the value of y ( t ) {\displaystyle y(t)} at discrete times t i {\displaystyle t_{i}} :

y i ≈ y ( t i ) where t i = t 0 + i h , {\displaystyle y_{i}\approx y(t_{i})\quad {\text{where}}\quad t_{i}=t_{0}+ih,}

where h {\displaystyle h} is the time step (sometimes referred to as Δ t {\displaystyle \Delta t} ) and i {\displaystyle i} is an integer. Multistep methods use information from the previous s {\displaystyle s} steps to calculate the next value. In particular, a linear multistep method uses a linear combination of y i {\displaystyle y_{i}} and f ( t i , y i ) {\displaystyle f(t_{i},y_{i})} to calculate the value of y {\displaystyle y} for the desired current step. Thus, a linear multistep method is a method of the form

y n + s + a s − 1 ⋅ y n + s − 1 + a s − 2 ⋅ y n + s − 2 + ⋯ + a 0 ⋅ y n

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Linear multistep method

Start with the simplest possible case. Write down what Linear multistep method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linear multistep method before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linear multistep method ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linear multistep method

In research: Linear multistep method appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Linear multistep method in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Linear multistep method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Numerical analysis, Numerical differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Linear multistep method outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Linear multistep method in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear multistep method means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Linear multistep method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear multistep method in simple terms?

Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point.

Why does Linear multistep method matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linear multistep method?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linear multistep method.