mathematics

Riemann solver

Riemann solver is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Riemann solver rather than just read about it. In short: A Riemann solver is a numerical method used to solve a hyperbolic partial differential equation based on the solution of the corresponding Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.

Key takeaways

Riemann 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 Riemann solver to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Riemann solver from memory before moving on to harder problems.

Reference excerpt

A Riemann solver is a numerical method used to solve a hyperbolic partial differential equation based on the solution of the corresponding Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.

Definition Generally speaking, Riemann solvers are specific methods for computing the numerical flux across a discontinuity in the Riemann problem. They form an important part of high-resolution schemes; typically the right and left states for the Riemann problem are calculated using some form of nonlinear reconstruction, such as a flux limiter or a WENO method, and then used as the input for the Riemann solver.

Exact solvers Sergei K. Godunov is credited with introducing the first exact Riemann solver for the Euler equations, by extending the previous CIR (Courant-Isaacson-Rees) method to non-linear systems of hyperbolic conservation laws. Modern solvers are able to simulate relativistic effects and magnetic fields. More recent research shows that an exact series solution to the Riemann problem exists, which may converge fast enough in some cases to avoid the iterative methods required in Godunov's scheme.

Approximate solvers As iterative solutions are too costly, especially in magnetohydrodynamics, some approximations have to be made. Some popular solvers are:

Roe solver

Philip L. Roe used the linearisation of the Jacobian, which he then solves exactly.

HLLE solver The HLLE solver (developed by Ami Harten, Peter Lax, Bram van Leer and B. Einfeldt) is an approximate solution to the Riemann problem, which is only based on the integral form of the conservation laws and the largest and smallest signal velocities at the interface. The stability and robustness of the HLLE solver is closely related to the signal velocities and a single central average state, as proposed by Einfeldt in the original paper

HLLC solver The HLLC (Harten-Lax-van Leer-Contact) solver was introduced by Toro. It restores the missing rarefaction wave by using an estimation technique, such as linearisation. More advanced techniques exist, like using the Roe average velocity for the middle wave speed. These schemes are quite robust and efficient but somewhat more diffusive.

Rotated-hybrid Riemann solvers These solvers were introduced by Hiroaki Nishikawa and Kitamura, in order to overcome the carbuncle problems of the Roe solver and the excessive diffusion of the HLLE solver at the same time. They developed robust and accurate Riemann solvers by combining the Roe solver and the HLLE/Rusanov solvers: they show that being applied in two orthogonal directions the two Riemann solvers can be combined into a single Roe-type solver (the Roe solver with modified wave speeds). In particular, the one derived from the Roe and HLLE solvers, called Rotated-RHLL solver, is extremely robust (carbuncle-free for all possible test cases on both structured and unstructured grids) and accurate (as accurate as the Roe solver for the boundary layer calculation).

Other solvers There are a variety of other solvers available, including more variants of the HLL scheme and solvers based on flux-splitting via characteristic decomposition.

Notes

See also Godunov's scheme Computational fluid dynamics Computational magnetohydrodynamics

References Toro, Eleuterio F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Berlin: Springer Verlag, ISBN 978-3-540-65966-2

External links

Illustrations

Worked examples

Example 1 — a first encounter with Riemann solver

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

In research: Riemann 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 Riemann 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: Riemann solver 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 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.

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How to study Riemann solver in 20 minutes

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

Frequently asked questions

What is Riemann solver in simple terms?

Why does Riemann 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 Riemann 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 Riemann solver.