Recursive collocation for the numerical solution of stiff ordinary differential equations

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Bibliographische Detailangaben
Zeitschriftentitel:	Mathematics of Computation
Personen und Körperschaften:	Brunner, H.
In:	Mathematics of Computation, 28, 1974, 126, S. 475-481
Format:	E-Article
Sprache:	Englisch
veröffentlicht:	American Mathematical Society (AMS)
Schlagwörter:	Applied Mathematics Computational Mathematics Algebra and Number Theory

Details
Zusammenfassung:	<p>The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval <italic>I</italic> is approximated, on each subinterval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponding to a partition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <italic>I</italic>, by a linear combination <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of exponential functions. The function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The unknown vectors in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are computed recursively by requiring that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy the given system at certain suitable points in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (collocation), with the additional condition that the collection of these functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper U Subscript k Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {U_k}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> represent a continuous function on <italic>I</italic> satisfying the given initial conditions.</p>
Umfang:	475-481
ISSN:	0025-5718 1088-6842
DOI:	10.1090/s0025-5718-1974-0347089-9