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Collocation for singular perturbation problems. II. Linear first order systems without turning points

Authors: U. Ascher and R. Weiss
Journal: Math. Comp. 43 (1984), 157-187
MSC: Primary 65L10; Secondary 34E15
MathSciNet review: 744929
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Abstract: We consider singularly perturbed linear boundary value problems for ODE's with variable coefficients, but without turning points. Convergence results are obtained for collocation schemes based on Gauss and Lobatto points, showing that highly accurate numerical solutions for these problems can be obtained at a very reasonable cost using such schemes, provided that appropriate meshes are used. The implementation of the numerical schemes and the practical construction of corresponding meshes are discussed.

These results extend those of a previous paper which deals with systems with constant coefficients.

References [Enhancements On Off] (What's this?)

  • [1] U. Ascher, S. Pruess & R. D. Russell, "On spline basis selection for solving differential equations," SIAM J. Numer. Anal, v. 20, 1983, pp. 121-142. MR 687372 (84h:65075)
  • [2] U. Ascher & R. Weiss, "Collocation for singular perturbation problems I: First order systems with constant coefficients," SIAM J. Numer. Anal., v. 20, 1983, pp. 537-557. MR 701095 (85a:65113)
  • [3] P. W. Hemker, A Numerical Study of Stiff Two-Point Boundary Value Problems, Math. Centrum, Amsterdam, 1977. MR 0488784 (58:8294)
  • [4] B. Kreiss & H. O. Kreiss, "Numerical methods for singular perturbation problems," SIAM J. Numer. Anal., v. 18, 1981, pp. 262-276. MR 612142 (82e:65088)
  • [5] H. O. Kreiss & N. Nichols, Numerical Methods for Singular Perturbation Problems, Dept. of Computer Science Report #57, Uppsala University, 1975. MR 0445849 (56:4182)
  • [6] P. A. Markowich & C. A. Ringhofer, "Collocation methods for boundary value problems on 'long' intervals," Math. Comp., v. 40, 1983, pp. 123-150. MR 679437 (84d:65053)
  • [7] R. D. Russell, "Collocation for systems of boundary value problems," Numer. Math. v. 23, 1974, 119-133. MR 0416074 (54:4150)
  • [8] R. Weiss, "The application of implicit Runge-Kutta and collocation methods to boundary-value problems," Math. Comp., v. 28, 1974, pp. 449-464. MR 0341881 (49:6627)
  • [9] R. Weiss, "An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems," Math. Comp., v. 42, 1984, pp. 41-67. MR 725984 (86b:65085)
  • [10] U. Ascher & R. Weiss, Collocation for Singular Perturbation Problems III: Nonlinear Problems Without Turning Points, Tech. Report 82-9, Univ. of British Columbia, Vancouver, Canada.
  • [11] C. de Boor & B. Swartz, "Collocation at Gaussian points," SIAM J. Numer. Anal., v. 10, 1973, pp. 582-606. MR 0373328 (51:9528)
  • [12] H. O. Kreiss, Centered Difference Approximation to Singular Systems of ODEs, Symposia Mathematica X (1972), Istituto Nazionale di Alta Matematica.
  • [13] P. Spudich & U. Ascher, "Calculation of complete theoretical seismograms in vertically varying media using collocation methods," Geoph. J. Roy. Astr. Soc., v. 75, 1983, pp. 101-124.

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Article copyright: © Copyright 1984 American Mathematical Society

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