# The ICICS/CS Reading Room

## UBC CS TR-81-02 Summary

- No on-line copy of this technical report is available.

- Collocation for Singular Perturbation Problems I: First Order Systems with Constant Coefficients, February 1981 Uri Ascher and R. Weiss
The application of collocation methods for the numerical solution of
singularly perturbed ordinary differential equations is investigated.
Collocation at Gauss, Radau and Lobatto points is considered, for both
initial and boundary value problems for first order systems with constant
coefficients. Particular attention is paid to symmetric schemes for boundary
value problems; these problems may have boundary layers at both interval
ends.
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Our analysis shows that certain collocation schemes, in particular
those based on Gauss or Lobatto points, do perform very well on such problems,
provided that a fine mesh with steps proportional to the layers' width is used
in the layers only, and a coarse mesh, just fine enough to resolve the
solution of the reduced problem, is used in between. Ways to construct
appropriate layer meshes are proposed. Of all methods considered, the Lobatto
schemes appear to be the most promising class of methods, as they essentially
retain their usual superconvergence power for the smooth, reduced solution,
whereas Gauss-Legendre schemes do not.
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We also investigate the conditioning of the linear systems of equations
arizing in the discretization of the boundary value problem. For a row
equilibrated version of the discretized system we obtain a pleasantly small
bound on the maximum norm condition number, which indicates that these systems
can be solved safely by Gaussian elimination with scaled partial pivoting.

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