Symmetric collocation methods for linear differential-algebraic boundary value problems |
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Authors: | Peter Kunkel Ronald Stöver |
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Institution: | 1.Fachbereich Mathematik, Carl von Ossietzky Universit?t, Postfach 2503, 26111 Oldenburg, Germany; e-mail: kunkel@math.uni-oldenburg.de
,DE;2.Universit?t Bremen, Fachbereich 3 - Mathematik und Informatik, Postfach 330 440, 28334 Bremen, Germany; e-mail: stoever@math.uni-bremen.de
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Abstract: | Summary. We present symmetric collocation methods for linear differential-algebraic boundary value problems without restrictions on
the index or the structure of the differential-algebraic equation. In particular, we do not require a separation into differential
and algebraic solution components. Instead, we use the splitting into differential and algebraic equations (which arises naturally
by index reduction techniques) and apply Gau?-type (for the differential part) and Lobatto-type (for the algebraic part) collocation
schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points. Under standard assumptions,
we show solvability and stability of the discrete problem and determine its order of convergence. Moreover, we show superconvergence
when using the combination of Gau? and Lobatto schemes and discuss the application of interpolation to reduce the number of
function evaluations. Finally, we present some numerical comparisons to show the reliability and efficiency of the new methods.
Received September 22, 2000 / Revised version received February 7, 2001 / Published online August 17, 2001 |
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Keywords: | Mathematics Subject Classification (1991): 65L10 |
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