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

Ascher, U.; Weiss, R.

doi:10.1090/S0025-5718-1984-0744929-2

Collocation for singular perturbation problems. II. Linear first order systems without turning points
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by U. Ascher and R. Weiss PDF

Math. Comp. 43 (1984), 157-187 Request permission

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

U. Ascher, S. Pruess, and R. D. Russell, On spline basis selection for solving differential equations, SIAM J. Numer. Anal. 20 (1983), no. 1, 121–142. MR 687372, DOI 10.1137/0720009
U. Ascher and R. Weiss, Collocation for singular perturbation problems. I. First order systems with constant coefficients, SIAM J. Numer. Anal. 20 (1983), no. 3, 537–557. MR 701095, DOI 10.1137/0720035
P. W. Hemker, A numerical study of stiff two-point boundary problems, Mathematical Centre Tracts, No. 80, Mathematisch Centrum, Amsterdam, 1977. MR 0488784
Barbro Kreiss and Heinz-Otto Kreiss, Numerical methods for singular perturbation problems, SIAM J. Numer. Anal. 18 (1981), no. 2, 262–276. MR 612142, DOI 10.1137/0718019
Heinz-Otto Kreiss and Nancy Nichols, Numerical methods for singular perturbation problems, Computing methods in applied sciences (Second Internat. Sympos.,Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976, pp. 544–558. MR 0445849
Peter A. Markowich and Christian A. Ringhofer, Collocation methods for boundary value problems on “long” intervals, Math. Comp. 40 (1983), no. 161, 123–150. MR 679437, DOI 10.1090/S0025-5718-1983-0679437-X
Robert D. Russell, Collocation for systems of boundary value problems, Numer. Math. 23 (1974), 119–133. MR 416074, DOI 10.1007/BF01459946
Richard Weiss, The application of implicit Runge-Kutta and collection methods to boundary-value problems, Math. Comp. 28 (1974), 449–464. MR 341881, DOI 10.1090/S0025-5718-1974-0341881-2
Richard Weiss, An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems, Math. Comp. 42 (1984), no. 165, 41–67. MR 725984, DOI 10.1090/S0025-5718-1984-0725984-2

Collocation for Singular Perturbation Problems

Nonlinear Problems Without Turning Points

Carl de Boor and Blâir Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582–606. MR 373328, DOI 10.1137/0710052

Centered Difference Approximation to Singular Systems of ODEs

Geoph. J. Roy. Astr. Soc.