Upwind scheme

Key takeaways

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

Reference excerpt

In computational physics, the term advection scheme refers to a class of numerical discretization methods for solving hyperbolic partial differential equations. In the so-called upwind schemes typically, the so-called upstream variables are used to calculate the derivatives in a flow field. That is, derivatives are estimated using a set of data points biased to be more "upwind" of the query point, with respect to the direction of the flow. Historically, the origin of upwind methods can be traced back to the work of Courant, Isaacson, and Rees who proposed the CIR method.

Model equation To illustrate the method, consider the following one-dimensional linear advection equation

∂ u ∂ t + a ∂ u ∂ x = 0 {\displaystyle {\frac {\partial u}{\partial t}}+a{\frac {\partial u}{\partial x}}=0}

which describes a wave propagating along the x {\displaystyle x} -axis with a velocity a {\displaystyle a} . This equation is also a mathematical model for one-dimensional linear advection. Consider a typical grid point i {\displaystyle i} in the domain. In a one-dimensional domain, there are only two directions associated with point i {\displaystyle i} – left (towards negative infinity) and right (towards positive infinity). If a {\displaystyle a} is positive, the traveling wave solution of the equation above propagates towards the right, the left side is called the upwind side and the right side is the downwind side. Similarly, if a {\displaystyle a} is negative the traveling wave solution propagates towards the left, the left side is called downwind side and right side is the upwind side. If the finite difference scheme for the spatial derivative, ∂ u / ∂ x {\displaystyle \partial u/\partial x} contains more points in the upwind side, the scheme is called an upwind-biased or simply an upwind scheme.

First-order upwind scheme

The simplest upwind scheme possible is the first-order upwind scheme. It is given by

where n {\displaystyle n} refers to the t {\displaystyle t} dimension and i {\displaystyle i} refers to the x {\displaystyle x} dimension. (By comparison, a central difference scheme in this scenario would look like

u i n + 1 − u i n Δ t + a u i + 1 n − u i − 1 n 2 Δ x = 0 , {\displaystyle {\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}+a{\frac {u_{i+1}^{n}-u_{i-1}^{n}}{2\Delta x}}=0,}

regardless of the sign of a {\displaystyle a} .)

Compact form Defining

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Upwind scheme

Start with the simplest possible case. Write down what Upwind scheme 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 Upwind scheme 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 Upwind scheme 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 Upwind scheme

In research: Upwind scheme 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 Upwind scheme 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: Upwind scheme is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational fluid dynamics, Numerical differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Upwind scheme 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 Upwind scheme in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Upwind scheme 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 Upwind scheme out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Upwind scheme in simple terms?

In computational physics, the term advection scheme refers to a class of numerical discretization methods for solving hyperbolic partial differential equations. In the so-called upwind schemes typically, the so-called upstream variables are used to calculate the derivatives in a flow field.

Why does Upwind scheme 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 Upwind scheme?

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 Upwind scheme.