clarke@utcsri.UUCP (Jim Clarke) (07/30/86)
(GB = Galbraith Building, 35 St. George Street)
NUMERICAL ANALYSIS SEMINAR, Tues. Aug. 5, 2 pm, GB248
Dr. Ivar Lie
NDRE, Kjeller, Norway
"Experiments with multistep collocation for stiff initial value"
The ideas behind multistep collocation methods are described briefly, fol-
lowed by the construction of the methods. This family of methods can be
formulated as multistep Runge-Kutta methods and the coefficients are com-
plex expressions of the collocation polynomial. For the k-step m-stage
methods, the order is m+k-1 by construction. Maximum attainable order is
2m+k-1 ordinate form. Since collocation methods are equivalent to fully
implicit Runge-Kutta methods one has to transform the RK-matrix to a
simpler form in order to implement them efficiently. A singly-implicit
form of the RK-matrix is convenient here, but other forms could also be
useful. Error estimation is done by imbedding the basic method in a par-
ticular multistep perturbed collocation method with m+1 stages. A singly-
implicit 2-step 2-stage method is constructed to form a basis for an exper-
imental code. This code has been compared to several well-known stiff in-
tegrators, e.g. EPISODE, SIMPLE and ROW4A. Performance results are given
for a set of stiff test problems. The multistep collocation code still
suffers from implementation deficiencies and is not yet as fast as the most
efficient stiff integrators.
Much work is still needed on theoretical aspects as well as on imple-
mentation before one can judge the usefulness of multistep collocation
methods. However, this family of methods seems to have certain potentials
in it, a new one being the use in construction of parallel multistep
Runge-Kutta methods.
NUMERICAL ANALYSIS SEMINAR, Thurs. Aug. 7, 2 pm, GB248
Dr. Stig Skelboe
University of Copenhagen
"Stability Properties of Linear Multirate Formulas"
--
Jim Clarke -- Dept. of Computer Science, Univ. of Toronto, Canada M5S 1A4
(416) 978-4058
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