Riemann problem

Key takeaways

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

Reference excerpt

A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest. The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations. In numerical analysis, Riemann problems appear in a natural way in finite volume methods for the solution of conservation law equations due to the discreteness of the grid. For that it is widely used in computational fluid dynamics and in computational magnetohydrodynamics simulations. In these fields, Riemann problems are calculated using Riemann solvers.

The Riemann problem in linearized gas dynamics As a simple example, we investigate the properties of the one-dimensional Riemann problem in gas dynamics (Toro, Eleuterio F. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics, Pg 44, Example 2.5) The initial conditions are given by

[ ρ u ] = [ ρ L u L ] for x ≤ 0 and [ ρ u ] = [ ρ R u R ] for x > 0 {\displaystyle {\begin{bmatrix}\rho \\u\end{bmatrix}}={\begin{bmatrix}\rho _{L}\\u_{L}\end{bmatrix}}{\text{ for }}x\leq 0\qquad {\text{and}}\qquad {\begin{bmatrix}\rho \\u\end{bmatrix}}={\begin{bmatrix}\rho _{R}\\u_{R}\end{bmatrix}}{\text{ for }}x>0}

where x = 0 separates two different states, together with the linearised gas dynamic equations (see gas dynamics for derivation).

∂ ρ ∂ t + ρ 0 ∂ u ∂ x = 0 ∂ u ∂ t + a 2 ρ 0 ∂ ρ ∂ x = 0 {\displaystyle {\begin{aligned}{\frac {\partial \rho }{\partial t}}+\rho _{0}{\frac {\partial u}{\partial x}}&=0\\[8pt]{\frac {\partial u}{\partial t}}+{\frac {a^{2}}{\rho _{0}}}{\frac {\partial \rho }{\partial x}}&=0\end{aligned}}}

where we can assume without loss of generality a ≥ 0 {\displaystyle a\geq 0} . We can now rewrite the above equations in a conservative form:

U t + A ⋅ U x = 0 {\displaystyle U_{t}+A\cdot U_{x}=0} : where

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Riemann problem

Start with the simplest possible case. Write down what Riemann problem 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 Riemann problem 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 Riemann problem 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 Riemann problem

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

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

Frequently asked questions

What is Riemann problem in simple terms?

A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest. The Riemann problem is very useful for the understanding of equations li…

Why does Riemann problem 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 Riemann problem?

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 Riemann problem.