Riemann problem

<p>A Riemann problem, named after <a href="/facts/Bernhard_Riemann/oV2fL0k8">Bernhard Riemann</a>, is a specific <a href="/facts/Initial_value_problem/PqmJfGML">initial value problem</a> composed of a  <a href="/facts/Conservation_law/6g1Ay2FR">conservation equation</a> together with <a href="/facts/Piecewise/nlQwptpe">piecewise</a> constant initial data which has a single <a href="/facts/Discontinuity_(mathematics)/TGLMsQom">discontinuity</a> in the domain of interest. The Riemann problem is very useful for the understanding of equations like <a href="/facts/Euler_equations_(fluid_dynamics)/EYW5lerl">Euler conservation equations</a> because all properties, such as shocks and rarefaction waves, appear as <a href="/facts/Method_of_characteristics/akP29vbW">characteristics</a> in the solution. It also gives an exact solution to some complex nonlinear equations, such as the <a href="/facts/Euler_equations_(fluid_dynamics)/EYW5lerl">Euler equations</a>.
</p><p>In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, Riemann problems appear in a natural way in <a href="/facts/Finite_volume_method/6JfQMucX">finite volume methods</a> for the solution of conservation law equations due to the discreteness of the grid. For that it is widely used in <a href="/facts/Computational_fluid_dynamics/zuWdtNJp">computational fluid dynamics</a> and in <a href="/facts/Computational_magnetohydrodynamics/m7B5t4fT">computational magnetohydrodynamics</a> simulations. In these fields, Riemann problems are calculated using <a href="/facts/Riemann_solver/3bY0sPc3">Riemann solvers</a>.
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Riemann problem open-in-new

Riemann problem