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<title>Riemann-Problem</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Riemann-Problem</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Als <b>Riemann-Problem</b> (nach <a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> (1826–1866)) wird in der <a href="Analysis" title="Analysis">Analysis</a> ein spezielles <a href="Anfangswertproblem" title="Anfangswertproblem">Anfangswertproblem</a> bezeichnet, bei dem die Anfangsdaten als konstant definiert werden, bis auf einen Punkt, in dem sie <a href="Unstetig" class="mw-redirect" title="Unstetig">unstetig</a> sind.

</p><p>Riemann-Probleme sind hilfreich für das Verständnis <a href="Hyperbolische_partielle_Differentialgleichung" class="mw-redirect" title="Hyperbolische partielle Differentialgleichung">hyperbolischer partieller Differentialgleichungen</a>, da in ihnen alle Phänomene wie <a href="Sto%C3%9Fwelle" title="Stoßwelle">Schocks</a>, <a href="Verdichtungssto%C3%9F" title="Verdichtungsstoß">Verdichtungsstöße</a> oder Verdünnungswellen auftauchen. Es sind auch für komplizierte nichtlineare Gleichungen wie die <a href="Eulersche_Gleichungen_(Str%C3%B6mungsmechanik)" class="mw-redirect" title="Eulersche Gleichungen (Strömungsmechanik)">Euler-Gleichungen der Strömungsmechanik</a> exakte Lösungen konstruierbar, was nicht für beliebige Anfangsdaten möglich ist.

</p><p>In der <a href="Numerische_Mathematik" title="Numerische Mathematik">numerischen Mathematik</a> tauchen Riemann-Probleme in natürlicher Weise in <a href="Finite-Volumen-Verfahren" title="Finite-Volumen-Verfahren">Finite-Volumen-Verfahren</a> zur Lösung von <a href="Erhaltungsgleichung" class="mw-redirect" title="Erhaltungsgleichung">Erhaltungsgleichungen</a> auf. Dort werden die Riemann-Probleme <a href="Approximation" title="Approximation">approximativ</a> mittels sogenannter Riemann-Löser angegangen.

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<div class="mw-heading mw-heading2"><h2 id="Erhaltungsgleichung">Erhaltungsgleichung</h2></div>

<p>Als wichtige <a href="Hyperbolische_partielle_Differentialgleichung" class="mw-redirect" title="Hyperbolische partielle Differentialgleichung">hyperbolische partielle Differentialgleichung</a> kann man Erhaltungsgleichungen des folgenden Typs betrachten:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\partial _{t}U+\partial _{x}F(U)&=0\\U(x,0)&=U_{0}(x)\end{aligned}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fcd93c26eee737a28f8c138657e6747c242593a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.054ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\partial _{t}U+\partial _{x}F(U)&=0\\U(x,0)&=U_{0}(x)\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Dabei gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\colon \mathbb {R} \times \mathbb {R} ^{+}\to \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle U\colon \mathbb {R} \times \mathbb {R} ^{+}\to \mathbb {R} ^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d35b4a50773b95b0e1d7e554f080c78f36679af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.034ex; height:2.509ex;" alt="{\displaystyle U\colon \mathbb {R} \times \mathbb {R} ^{+}\to \mathbb {R} ^{n}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c13fd91a344a58a3e65cbdb662518a5979b75c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.182ex; height:2.343ex;" alt="{\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}" loading="lazy"></span>.

</p><p>Beim Riemann-Problem gilt für den Anfangswert:

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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}U_{0}(x)={\begin{cases}U_{L},\quad x<0\\U_{R},\quad x>0\end{cases}}\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U_{0}(x)={\begin{cases}U_{L},\quad x<0\\U_{R},\quad x>0\end{cases}}\end{aligned}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aff4eee74b6a5fd7d3d8c9e313d09eb358576bd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.139ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}U_{0}(x)={\begin{cases}U_{L},\quad x<0\\U_{R},\quad x>0\end{cases}}\end{aligned}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle U_{L},U_{R}\in \mathbb {R} ^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9b3c875280f1914823b6dc5f7d39a1ba470ad31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.777ex; height:2.676ex;" alt="{\displaystyle U_{L},U_{R}\in \mathbb {R} ^{n}}" loading="lazy"></span>.

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<div class="mw-heading mw-heading3"><h3 id="Linearer_Fluss">Linearer Fluss</h3></div>

<p>Für den linearen Fluss

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(U)=AU,\quad A\in \mathbb {R} ^{n\times n}\end{aligned}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54ff842b81e05c2f7b08b359a71f4de2d2ed149b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.81ex; height:2.843ex;" alt="{\displaystyle {\begin{aligned}F(U)=AU,\quad A\in \mathbb {R} ^{n\times n}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>lässt sich die analytische Lösung berechnen.

Für ein hyperbolisches Problem ist die Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> stets diagonalisierbar:

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<annotation encoding="application/x-tex">{\displaystyle TAT^{-1}=\Lambda =\operatorname {diag} (\lambda _{1},\dotsc ,\lambda _{n})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/274f26b847bea1c1aec82d226e39eb54666c3640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.477ex; height:3.176ex;" alt="{\displaystyle TAT^{-1}=\Lambda =\operatorname {diag} (\lambda _{1},\dotsc ,\lambda _{n})}" loading="lazy"></span></dd></dl>

<p>mit einer Basistransformationsmatrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in \mathbb {R} ^{n\times n}}">

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<annotation encoding="application/x-tex">{\displaystyle T\in \mathbb {R} ^{n\times n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bd17d7bbc0788eb732fdd7f6e60dd79b62f63e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.638ex; height:2.343ex;" alt="{\displaystyle T\in \mathbb {R} ^{n\times n}}" loading="lazy"></span>.

</p><p>Mit der Transformation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W:=T^{-1}U}">

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<annotation encoding="application/x-tex">{\displaystyle W:=T^{-1}U}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/574e2b54f498b35e1ca603f82bc8eaad6448525f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.016ex; height:2.676ex;" alt="{\displaystyle W:=T^{-1}U}" loading="lazy"></span> kann man die <a href="PDGL" class="mw-redirect" title="PDGL">PDGL</a> entkoppeln:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8d0be2974c2625dd4da281730669ef2793578c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:42.425ex; height:12.509ex;" alt="{\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}}}" loading="lazy"></span></dd></dl>

<p>Entkopplung bedeutet in diesem Fall, dass in der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}" loading="lazy"></span> vorkommen.

</p><p>Jede einzelne Gleichung entspricht einer <a href="Partielle_Differentialgleichung#Einführung" title="Partielle Differentialgleichung">linearen, skalaren Transportgleichung</a> und somit ist die Lösung einfach zu bestimmen:

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<annotation encoding="application/x-tex">{\displaystyle W_{i}(x,t)=(W_{0})_{i}(x-\lambda _{i}t).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c0cee8fa66ead235c65d9afe2c7713115dec8eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.582ex; height:2.843ex;" alt="{\displaystyle W_{i}(x,t)=(W_{0})_{i}(x-\lambda _{i}t).}" loading="lazy"></span></dd></dl>

<p>Rücktransformation ergibt nun die gesuchte Lösung:

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<annotation encoding="application/x-tex">{\displaystyle U(x,t)=TW(x,t).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0abc0a74d5a48cc830bc7a180243a61492e7a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.624ex; height:2.843ex;" alt="{\displaystyle U(x,t)=TW(x,t).}" loading="lazy"></span></dd></dl>

<p>Man kann die Lösung auch anders erhalten, indem man den Sprung der Anfangswerte in der neuen Basis darstellt:

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<annotation encoding="application/x-tex">{\displaystyle U_{R}-U_{L}=\sum _{j=1}^{n}\alpha _{j}t_{j}\quad {\text{mit }}\alpha _{j}\in \mathbb {R} ,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/852a768bf5739c5a1ea5314c8fda701ffe00604f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.787ex; height:7.176ex;" alt="{\displaystyle U_{R}-U_{L}=\sum _{j=1}^{n}\alpha _{j}t_{j}\quad {\text{mit }}\alpha _{j}\in \mathbb {R} ,}" loading="lazy"></span></dd></dl>

<p>wobei die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{j}\in \mathbb {R} ^{n}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43767cedb2f1444f2daa2d1cc5a68fe2174d53a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.487ex; height:3.009ex;" alt="{\displaystyle t_{j}\in \mathbb {R} ^{n}}" loading="lazy"></span> die Eigenvektoren von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> sind (also: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=(t_{1},\dotsc ,t_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle T=(t_{1},\dotsc ,t_{n})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2105bb22ba4f4b5bc9c531b78863df7c2a5c5c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.674ex; height:2.843ex;" alt="{\displaystyle T=(t_{1},\dotsc ,t_{n})}" loading="lazy"></span>).

Nun ist die Lösung so gegeben:

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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x,t)=U_{L}+\sum _{\lambda _{j}<{\frac {x}{t}}}\alpha _{j}t_{j}=U_{R}-\sum _{\lambda _{j}>{\frac {x}{t}}}\alpha _{j}t_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle U(x,t)=U_{L}+\sum _{\lambda _{j}<{\frac {x}{t}}}\alpha _{j}t_{j}=U_{R}-\sum _{\lambda _{j}>{\frac {x}{t}}}\alpha _{j}t_{j}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe479bf21a881a15c9d934a361c7d8b1964a4e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:42.848ex; height:6.843ex;" alt="{\displaystyle U(x,t)=U_{L}+\sum _{\lambda _{j}<{\frac {x}{t}}}\alpha _{j}t_{j}=U_{R}-\sum _{\lambda _{j}>{\frac {x}{t}}}\alpha _{j}t_{j}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Eleuterio F. Toro: <i>Riemann Solvers and Numerical Methods for Fluid Dynamics</i>, Springer Verlag, Berlin 1999, ISBN 3-540-65966-8.</li>

<li>Randall J. LeVeque: <i>Finite-Volume Methods for Hyperbolic Problems</i>, Cambridge University Press, Cambridge 2004, ISBN 0-521-81087-6.</li></ul></div><div><div class="zim-footer">

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