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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Trapez-Methode</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>Das implizite <b>Trapez-Verfahren</b> ist ein Verfahren zur <a href="Numerische_Mathematik" title="Numerische Mathematik">numerischen</a> Lösung eines <a href="Anfangswertproblem" title="Anfangswertproblem">Anfangswertproblems</a>
</p>
<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 y'(t)=f\left(t,y(t)\right),\quad y(t_{0})=y_{0}}">
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<annotation encoding="application/x-tex">{\displaystyle y'(t)=f\left(t,y(t)\right),\quad y(t_{0})=y_{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/376e80039eb1ba8f428e4fdec157e6a4bdbf055f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.64ex; height:3.009ex;" alt="{\displaystyle y'(t)=f\left(t,y(t)\right),\quad y(t_{0})=y_{0}}" loading="lazy"></span></dd></dl>
<p>Es lässt sich sowohl den <a href="Runge-Kutta-Verfahren" title="Runge-Kutta-Verfahren">Runge-Kutta-Verfahren</a> als auch den <a href="Mehrschrittverfahren#Implizite_Verfahren" title="Mehrschrittverfahren">Adams-Moulton-Verfahren</a> zuordnen. Das Trapezverfahren ist <a href="A-Stabilit%C3%A4t" title="A-Stabilität">A-stabil</a> mit der Besonderheit, dass für die Schwingungsgleichung <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 y'=\mathrm {i} \alpha y}">
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<annotation encoding="application/x-tex">{\displaystyle y'=\mathrm {i} \alpha y}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/094dee4b363bb1e8e5507c0d65541b9d8b476ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.234ex; height:2.843ex;" alt="{\displaystyle y'=\mathrm {i} \alpha y}" loading="lazy"></span> kein Amplitudenfehler auftritt<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. Das Verfahren lässt sich aus der <a href="Trapezregel" title="Trapezregel">Trapezregel</a> herleiten:
</p>
<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 y_{n+1}=y_{n}+{\frac {h}{2}}(f_{n+1}+f_{n})}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle y_{n+1}=y_{n}+{\frac {h}{2}}(f_{n+1}+f_{n})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3092c46f392a55844e149cf1780fe8375e16c4ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.396ex; height:5.343ex;" alt="{\displaystyle y_{n+1}=y_{n}+{\frac {h}{2}}(f_{n+1}+f_{n})}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<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 f_{n}\ :=f(t_{n},y_{n}).}">
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<annotation encoding="application/x-tex">{\displaystyle f_{n}\ :=f(t_{n},y_{n}).}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b5d1db20cafde28432e4439e4e938b08eea4a7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.868ex; height:2.843ex;" alt="{\displaystyle f_{n}\ :=f(t_{n},y_{n}).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Herleitung">Herleitung</h2></div>
<p>Für die Herleitung von <a href="Einschrittverfahren" title="Einschrittverfahren">Einschrittverfahren</a> wird das <a href="Anfangswertproblem" title="Anfangswertproblem">Anfangswertproblem</a> meist in der zu ihr äquivalenten <a href="Integralgleichung" title="Integralgleichung">Integralgleichung</a> umgeformt<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<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}{\dot {y}}&=f(t,y)\,,\qquad y(t_{0})=y_{0}\\\Longleftrightarrow \quad y(t)&=y_{0}+\int _{t_{0}}^{t}f(s,y(s))\,\mathrm {d} s\,.\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\dot {y}}&=f(t,y)\,,\qquad y(t_{0})=y_{0}\\\Longleftrightarrow \quad y(t)&=y_{0}+\int _{t_{0}}^{t}f(s,y(s))\,\mathrm {d} s\,.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14f97cae7004afe108e0fe5fb63582dab64573a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.988ex; margin-bottom: -0.183ex; width:37.272ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}{\dot {y}}&=f(t,y)\,,\qquad y(t_{0})=y_{0}\\\Longleftrightarrow \quad y(t)&=y_{0}+\int _{t_{0}}^{t}f(s,y(s))\,\mathrm {d} s\,.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Nun besteht die Idee bei der <i>impliziten Trapez-Methode</i> eine simple <a href="Numerische_Integration" title="Numerische Integration">Quadraturformel</a> für das Integral zu benutzen: die <a href="Trapezregel#Sehnentrapezformel" title="Trapezregel">Trapezregel</a>. Man approximiert in jedem <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 k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-ten Schritt den Integranden wie folgt
</p>
<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 \int _{t_{k}}^{t_{k+1}}f(s,y(s))\,\mathrm {d} s\approx {\frac {h}{2}}{\Big (}f(t_{k},y_{k})+f(t_{k+1},y_{k+1}){\Big )}\,.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \int _{t_{k}}^{t_{k+1}}f(s,y(s))\,\mathrm {d} s\approx {\frac {h}{2}}{\Big (}f(t_{k},y_{k})+f(t_{k+1},y_{k+1}){\Big )}\,.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcffe94eba30ba0881d42c1b37ce21c5f98172df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.96ex; height:6.509ex;" alt="{\displaystyle \int _{t_{k}}^{t_{k+1}}f(s,y(s))\,\mathrm {d} s\approx {\frac {h}{2}}{\Big (}f(t_{k},y_{k})+f(t_{k+1},y_{k+1}){\Big )}\,.}" loading="lazy"></span></dd></dl>
<p>Zusammen ergibt dies die <b>Trapez-Methode</b><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<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 y(t_{k+1})=y(t_{k})+\int _{t_{k}}^{t_{k+1}}f(s,y(s))\,\mathrm {d} s\approx y_{k}+{\frac {h}{2}}{\Big (}f(t_{k},y_{k})+f(t_{k+1},y_{k+1}){\Big )}=:y_{k+1}\,.}">
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<mi>y</mi>
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<mi>s</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>s</mi>
<mo>≈<!-- ≈ --></mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.623em" minsize="1.623em">(</mo>
</mrow>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.623em" minsize="1.623em">)</mo>
</mrow>
</mrow>
<mo>=:</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y(t_{k+1})=y(t_{k})+\int _{t_{k}}^{t_{k+1}}f(s,y(s))\,\mathrm {d} s\approx y_{k}+{\frac {h}{2}}{\Big (}f(t_{k},y_{k})+f(t_{k+1},y_{k+1}){\Big )}=:y_{k+1}\,.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92c98848b645a9f98c59a27c1ea1c3f7ff6731c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:81.928ex; height:6.509ex;" alt="{\displaystyle y(t_{k+1})=y(t_{k})+\int _{t_{k}}^{t_{k+1}}f(s,y(s))\,\mathrm {d} s\approx y_{k}+{\frac {h}{2}}{\Big (}f(t_{k},y_{k})+f(t_{k+1},y_{k+1}){\Big )}=:y_{k+1}\,.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Lösungsmethode"><span id="L.C3.B6sungsmethode"></span>Lösungsmethode</h2></div>
<p>Zur Lösung dieses, in der Regel nichtlinearen, Gleichungssystems können verschiedene numerische Verfahren genutzt werden. Für das quadratisch konvergente <a href="Newton-Verfahren" class="mw-redirect" title="Newton-Verfahren">Newton-Verfahren</a> ergibt sich konkret:
</p>
<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 y_{n+1}^{(k+1)}=y_{n+1}^{(k)}-\left(I-{\frac {h}{2}}{\frac {\partial f_{n+1}^{(k)}}{\partial y_{n+1}^{(k)}}}\right)^{-1}\left(y_{n+1}^{(k)}-y_{n}-{\frac {h}{2}}(f_{n+1}^{(k)}+f_{n})\right).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo>−<!-- − --></mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>I</mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msubsup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo>−<!-- − --></mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<msubsup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo>+</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{n+1}^{(k+1)}=y_{n+1}^{(k)}-\left(I-{\frac {h}{2}}{\frac {\partial f_{n+1}^{(k)}}{\partial y_{n+1}^{(k)}}}\right)^{-1}\left(y_{n+1}^{(k)}-y_{n}-{\frac {h}{2}}(f_{n+1}^{(k)}+f_{n})\right).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01c9a9083204de61b684a446142d78a06f438c3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:66.205ex; height:8.843ex;" alt="{\displaystyle y_{n+1}^{(k+1)}=y_{n+1}^{(k)}-\left(I-{\frac {h}{2}}{\frac {\partial f_{n+1}^{(k)}}{\partial y_{n+1}^{(k)}}}\right)^{-1}\left(y_{n+1}^{(k)}-y_{n}-{\frac {h}{2}}(f_{n+1}^{(k)}+f_{n})\right).}" loading="lazy"></span></dd></dl>
<p>Man erhält also ein <a href="Lineares_Gleichungssystem" title="Lineares Gleichungssystem">lineares Gleichungssystem</a>
</p>
<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 (I-{\frac {h}{2}}J^{(k)})y_{n+1}^{(k+1)}=-{\frac {h}{2}}J^{(k)}y_{n+1}^{(k)}+y_{n}+{\frac {h}{2}}(f_{n+1}^{(k)}+f_{n}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo>+</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<msubsup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mo>+</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (I-{\frac {h}{2}}J^{(k)})y_{n+1}^{(k+1)}=-{\frac {h}{2}}J^{(k)}y_{n+1}^{(k)}+y_{n}+{\frac {h}{2}}(f_{n+1}^{(k)}+f_{n}),}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4d0076a9e9de9a2fd2c52f4d7b352860e15f768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.279ex; height:5.343ex;" alt="{\displaystyle (I-{\frac {h}{2}}J^{(k)})y_{n+1}^{(k+1)}=-{\frac {h}{2}}J^{(k)}y_{n+1}^{(k)}+y_{n}+{\frac {h}{2}}(f_{n+1}^{(k)}+f_{n}),}" loading="lazy"></span></dd></dl>
<p>wobei <i>J</i> die <a href="Jacobi-Matrix" title="Jacobi-Matrix">Jacobi-Matrix</a>
</p>
<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 J^{(k)}:=\left({\frac {\partial f}{\partial y}}\right)_{n+1}^{(k)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>:=</mo>
<msubsup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J^{(k)}:=\left({\frac {\partial f}{\partial y}}\right)_{n+1}^{(k)}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d060efbc3450e674c59037a6a31a3944878c8e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.812ex; height:6.843ex;" alt="{\displaystyle J^{(k)}:=\left({\frac {\partial f}{\partial y}}\right)_{n+1}^{(k)}}" loading="lazy"></span>,</dd></dl>
<p><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> die <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a> 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 k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> der Iterationsschritt ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Stabilität"><span id="Stabilit.C3.A4t"></span>Stabilität</h2></div>
<p>Mit der Testgleichung <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 y'(t)=\lambda y(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>y</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>λ<!-- λ --></mi>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y'(t)=\lambda y(t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe31273818b08bcaa7b7fd86a888612c3c552f6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.752ex; height:3.009ex;" alt="{\displaystyle y'(t)=\lambda y(t)}" loading="lazy"></span> bekommt man die <a href="Stabilit%C3%A4tsfunktion" title="Stabilitätsfunktion">Stabilitätsfunktion</a>
</p>
<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 R(z)={\frac {2+z}{2-z}},\quad z=h\lambda \in \mathbb {C} .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mi>z</mi>
</mrow>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mi>z</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mspace width="1em"></mspace>
<mi>z</mi>
<mo>=</mo>
<mi>h</mi>
<mi>λ<!-- λ --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R(z)={\frac {2+z}{2-z}},\quad z=h\lambda \in \mathbb {C} .}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4536dfb2112bf5f116bb5245f933d9c46aafb5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.09ex; height:5.343ex;" alt="{\displaystyle R(z)={\frac {2+z}{2-z}},\quad z=h\lambda \in \mathbb {C} .}" loading="lazy"></span></dd></dl>
<p>Auf der imaginären Achse <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 z=i\eta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mi>i</mi>
<mi>η<!-- η --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=i\eta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/876b74811ae3cb322dc3e04f6ff69e632fe04926.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.158ex; height:2.676ex;" alt="{\displaystyle z=i\eta }" loading="lazy"></span> 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 |R(i\eta )|=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mi>η<!-- η --></mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |R(i\eta )|=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4890490d2f763e9b4ccd5c4bd61eeef2e7364b72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.1ex; height:2.843ex;" alt="{\displaystyle |R(i\eta )|=1}" loading="lazy"></span>, daher ist die Trapezmethode <a href="A-Stabilit%C3%A4t" title="A-Stabilität">A-stabil</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Schrittweite_h">Schrittweite <i>h</i></h2></div>
<p>Die (variable) Schrittweite kann aus folgender Beziehung berechnet werden:
</p>
<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 \left\vert {\frac {R(h\lambda )}{\mathrm {e} ^{h\lambda }}}-1\right\vert =\delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>|</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>h</mi>
<mi>λ<!-- λ --></mi>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\vert {\frac {R(h\lambda )}{\mathrm {e} ^{h\lambda }}}-1\right\vert =\delta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b61560be26b22a6add213a063c71776d9f2f585.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.547ex; height:6.509ex;" alt="{\displaystyle \left\vert {\frac {R(h\lambda )}{\mathrm {e} ^{h\lambda }}}-1\right\vert =\delta }" loading="lazy"></span>;</dd></dl>
<p><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 \delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span> bezeichnet den zugelassenen <a href="Lokaler_Diskretisierungsfehler" title="Lokaler Diskretisierungsfehler">lokalen Diskretisierungsfehler</a>. Der Ansatz <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 y_{n+1}=y_{n}+{\frac {h}{2}}(f_{n+1}+f_{n})=:R(h\lambda )y_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=:</mo>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{n+1}=y_{n}+{\frac {h}{2}}(f_{n+1}+f_{n})=:R(h\lambda )y_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fda00a65c037fb4dc298c60fd8dc6e8c90c0fdb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.766ex; height:5.343ex;" alt="{\displaystyle y_{n+1}=y_{n}+{\frac {h}{2}}(f_{n+1}+f_{n})=:R(h\lambda )y_{n}}" loading="lazy"></span> liefert für die implizite Trapez-Methode
</p>
<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 R(h\lambda )={\frac {2+h\lambda }{2-h\lambda }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mi>h</mi>
<mi>λ<!-- λ --></mi>
</mrow>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mi>h</mi>
<mi>λ<!-- λ --></mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R(h\lambda )={\frac {2+h\lambda }{2-h\lambda }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/710ebbb073bbfd33209523e04bdf6db261bc1c0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.899ex; height:5.676ex;" alt="{\displaystyle R(h\lambda )={\frac {2+h\lambda }{2-h\lambda }}}" loading="lazy"></span>.</dd></dl>
<p>Dabei ist <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 \lambda \,:=\max _{j}{|\lambda _{j}|}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mspace width="thinmathspace"></mspace>
<mo>:=</mo>
<munder>
<mo movablelimits="true" form="prefix">max</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda \,:=\max _{j}{|\lambda _{j}|}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b1bbd7a117f036016f54e758df325d8f684f021.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.759ex; height:4.343ex;" alt="{\displaystyle \lambda \,:=\max _{j}{|\lambda _{j}|}}" loading="lazy"></span> der Betrag des betragsmäßig größten Eigenwerts der Jacobi-Matrix (<a href="Spektralradius" title="Spektralradius">Spektralradius</a>). Die numerische Bestimmung der Eigenwerte ist sehr zeitaufwendig; für den Zweck der Schrittweitenberechnung ist es im Allgemeinen ausreichend die <a href="Gesamtnorm" title="Gesamtnorm">Gesamtnorm</a> <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 \lambda =N\cdot \max _{i,j}|a_{ij}|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mo>=</mo>
<mi>N</mi>
<mo>⋅<!-- ⋅ --></mo>
<munder>
<mo movablelimits="true" form="prefix">max</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda =N\cdot \max _{i,j}|a_{ij}|}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63765eae888cb3c4e32e1977e8139b76b7aae4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.91ex; height:4.343ex;" alt="{\displaystyle \lambda =N\cdot \max _{i,j}|a_{ij}|}" loading="lazy"></span> heranzuziehen, die immer größer oder gleich der <a href="Spektralnorm" title="Spektralnorm">Spektralnorm</a> ist. <i>N</i> ist der Rang der Jacobi-Matrix 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 a_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebea6cd2813c330c798921a2894b358f7b643917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}" loading="lazy"></span> deren Elemente.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Hans R. Schwarz, Norbert Köckler: <i>Numerische Mathematik.</i> 5. Auflage, Teubner, Stuttgart 2004, ISBN 3-519-42960-8, S. 343.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">M. Kloker: <a rel="nofollow" class="external text" href="https://www.iag.uni-stuttgart.de/lehre/lehrveranstaltungen/lehrveranst_master/einz_vorl_master/pdf_transition_kloker/vnum_katalog_1.pdf"><i>Numerische Löser (Zeitintegrationsverfahren) für die Gewöhnliche Modelldifferentialgleichung y'=αy</i></a> (PDF; 2,2 MB), Universität Stuttgart, 1996</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Reusken, Arnold.: <cite style="font-style:italic">Numerik für Ingenieure und Naturwissenschaftler</cite>. Springer, Berlin 2006, ISBN 3-540-25544-3, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>378</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Trapez-Methode&rft.au=Reusken%2C+Arnold.&rft.btitle=Numerik+f%C3%BCr+Ingenieure+und+Naturwissenschaftler&rft.date=2006&rft.genre=book&rft.isbn=3540255443&rft.pages=378&rft.place=Berlin&rft.pub=Springer" style="display:none"> </span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Reusken, Arnold.: <cite style="font-style:italic">Numerik für Ingenieure und Naturwissenschaftler</cite>. Springer, Berlin 2006, ISBN 3-540-25544-3, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>383</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Trapez-Methode&rft.au=Reusken%2C+Arnold.&rft.btitle=Numerik+f%C3%BCr+Ingenieure+und+Naturwissenschaftler&rft.date=2006&rft.genre=book&rft.isbn=3540255443&rft.pages=383&rft.place=Berlin&rft.pub=Springer" style="display:none"> </span></span>
</li>
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