In analysis, the Riemann problem (after Bernhard Riemann (1826-1866)) is a special initial value problem in which the initial data are defined as constant, except for a point at which they are discontinuous .
Riemann problems are very helpful for understanding hyperbolic partial differential equations , as all phenomena such as shocks , shock waves or dilution waves appear in them. Exact solutions can also be constructed for complicated non-linear equations such as the Euler equations in fluid mechanics , which is not possible for any initial data.
In numerical mathematics , Riemann problems appear naturally in finite volume methods for solving conservation equations . There the Riemann problems are approached approximately using so-called Riemann solvers .
Conservation equation in nD
As an important hyperbolic partial differential equation one can consider conservation equations of the following type:
where and applies.
In the Riemann problem, the following applies to the initial value:
For the following linear flow:
the analytical solution can be calculated. For hyperbolic problems the matrix is always diagonalizable:
with a base transformation matrix .
The transformation can be used to decouple the PDE:
In this case, decoupling means that the PDE line only contains derivatives of .
Each individual equation corresponds to a linear, scalar transport equation and thus the solution is easy to determine:
Inverse transformation now gives the solution:
The solution can also be obtained differently by showing the jump of the initial values in the new basis:
where are the eigenvectors of (i.e.:) . Now the solution is given as:
- Eleuterio F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics , Springer Verlag, Berlin 1999, ISBN 3-540-65966-8 .
- Randall J. LeVeque: Finite-Volume Methods for Hyperbolic Problems , Cambridge University Press, Cambridge 2004, ISBN 0-521-81087-6 .