Riemann problem

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:

${\displaystyle {\begin{aligned}\partial _{t}U+\partial _{x}F(U)&=0\\U(x,0)&=U_{0}(x)\end{aligned}}}$

where and applies. ${\displaystyle U\colon \mathbb {R} \times \mathbb {R} ^{+}\to \mathbb {R} ^{n}}$ ${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$

In the Riemann problem, the following applies to the initial value:

${\displaystyle {\begin{aligned}U_{0}(x)={\begin{cases}U_{L}\quad ,x<0\\U_{R}\quad ,x>0\end{cases}}\end{aligned}}}$

for . ${\displaystyle U_{L},U_{R}\in \mathbb {R} ^{n}}$

Linear flow

For the following linear flow:

${\displaystyle {\begin{aligned}F(U)=AU,\quad A\in \mathbb {R} ^{n\times n}\end{aligned}}}$

the analytical solution can be calculated. For hyperbolic problems the matrix is always diagonalizable: ${\displaystyle A}$

${\displaystyle TAT^{-1}=\Lambda =\operatorname {diag} (\lambda _{1},\dotsc ,\lambda _{n})}$

with a base transformation matrix . ${\displaystyle T\in \mathbb {R} ^{n\times n}}$

The transformation can be used to decouple the PDE: ${\displaystyle W:=T^{-1}U}$

${\displaystyle {\begin{aligned}&\left\lbrace {\begin{aligned}\partial _{t}U+A\partial _{x}U&=0\\U(x,0)&=U_{0}(x)\end{aligned}}\right.\\\Leftrightarrow &\left\lbrace {\begin{aligned}\partial _{t}W+\Lambda \partial _{x}W&=0\\W(x,0)&=W_{0}(x):=T^{-1}U_{0}(x)\end{aligned}}\right.\end{aligned}}}$

In this case, decoupling means that the PDE line only contains derivatives of . ${\displaystyle i.}$ ${\displaystyle W_{i}}$

Each individual equation corresponds to a linear, scalar transport equation and thus the solution is easy to determine:

${\displaystyle W_{i}(x,t)=(W_{0})_{i}(x-\lambda _{i}t).}$

Inverse transformation now gives the solution:

${\displaystyle U(x,t)=TW(x,t).}$

The solution can also be obtained differently by showing the jump of the initial values ​​in the new basis:

${\displaystyle U_{R}-U_{L}=\sum _{j=1}^{n}\alpha _{j}t_{j}\quad {\text{mit }}\alpha _{j}\in \mathbb {R} ,}$

where are the eigenvectors of (i.e.:) . Now the solution is given as: ${\displaystyle t_{j}\in \mathbb {R} ^{n}}$ ${\displaystyle A}$ ${\displaystyle T=(t_{1},\dotsc ,t_{n})}$

${\displaystyle U(x,t)=U_{L}+\sum _{j:{\frac {x}{t}}>\lambda _{j}}\alpha _{j}t_{j}=U_{R}-\sum _{j:{\frac {x}{t}}<\lambda _{j}}\alpha _{j}t_{j}.}$

literature

• 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 .