Roe solver

Key takeaways

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

Reference excerpt

The Roe approximate Riemann solver, devised by Phil Roe, is an approximate Riemann solver based on the Godunov scheme and involves finding an estimate for the intercell numerical flux or Godunov flux F i + 1 2 {\displaystyle F_{i+{\frac {1}{2}}}} at the interface between two computational cells U i {\displaystyle U_{i}} and U i + 1 {\displaystyle U_{i+1}} , on some discretised space-time computational domain.

Roe scheme

Quasi-linear hyperbolic system A non-linear system of hyperbolic partial differential equations representing a set of conservation laws in one spatial dimension can be written in the form

∂ U ∂ t + ∂ F ( U ) ∂ x = 0. {\displaystyle {\frac {\partial {\boldsymbol {U}}}{\partial t}}+{\frac {\partial {\boldsymbol {F}}({\boldsymbol {U}})}{\partial x}}=0.}

Applying the chain rule to the second term we get the quasi-linear hyperbolic system

∂ U ∂ t + A ( U ) ∂ U ∂ x = 0 , {\displaystyle {\frac {\partial {\boldsymbol {U}}}{\partial t}}+A({\boldsymbol {U}}){\frac {\partial {\boldsymbol {U}}}{\partial x}}=0,}

where A {\displaystyle A} is the Jacobian matrix of the flux vector F ( U ) {\displaystyle {\boldsymbol {F}}({\boldsymbol {U}})} .

Roe matrix The Roe method consists of finding a matrix A ~ ( U i , U i + 1 ) {\displaystyle {\tilde {A}}({\boldsymbol {U}}_{i},{\boldsymbol {U}}_{i+1})} that is assumed constant between two cells. The Riemann problem can then be solved as a truly linear hyperbolic system at each cell interface. The Roe matrix must obey the following conditions:

Diagonalizable with real eigenvalues: ensures that the new linear system is truly hyperbolic. Consistency with the exact jacobian: when U i , U i + 1 → U {\displaystyle {\boldsymbol {U}}_{i},{\boldsymbol {U}}_{i+1}\rightarrow {\boldsymbol {U}}} we demand that A ~ ( U i , U i + 1 ) = A ( U ) {\displaystyle {\tilde {A}}({\boldsymbol {U}}_{i},{\boldsymbol {U}}_{i+1})=A({\boldsymbol {U}})}

Conserving: F i + 1 − F i = A ~ ( U i + 1 − U i ) {\displaystyle {\boldsymbol {F}}_{i+1}-{\boldsymbol {F}}_{i}={\tilde {A}}({\boldsymbol {U}}_{i+1}-{\boldsymbol {U}}_{i})}

Phil Roe introduced a method of parameter vectors to find such a matrix for some systems of conservation laws.

Intercell flux Once the Roe matrix corresponding to the interface between two cells is found, the intercell flux is given by solving the quasi-linear system as a truly linear system.

Worked examples

Example 1 — a first encounter with Roe solver

Start with the simplest possible case. Write down what Roe solver 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 Roe solver 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 Roe solver 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 Roe solver

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

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

Frequently asked questions

What is Roe solver in simple terms?

The Roe approximate Riemann solver, devised by Phil Roe, is an approximate Riemann solver based on the Godunov scheme and involves finding an estimate for the intercell numerical flux or Godunov flux F i + 1 2 {\displaystyle F_{i+{\frac {1}{2}}}} at the interface between two computational cells U i…

Why does Roe solver 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 Roe solver?

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 Roe solver.