Application of method of collocation on lines for solving nonlinear hyperbolic problems

Author:
E. N. Houstis

Journal:
Math. Comp. **31** (1977), 443-456

MSC:
Primary 65N35

MathSciNet review:
0443379

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Abstract | References | Similar Articles | Additional Information

Abstract: A collocation on lines procedure based on piecewise polynomials is applied to initial/boundary value problems for nonlinear hyperbolic partial differential equations. Optimal order a priori estimates are obtained for the error of approximation. The Crank-Nicholson discretization in time is studied and convergence rates of the collocation-Crank-Nicholson procedure are established. Finally, the superconvergence is verified at particular points for linear hyperbolic problems.

**[1]**I. S. Berezin and N. P. Shidkov,*Computing methods. Vols. I, II*, Translated by O. M. Blunn; translation edited by A. D. Booth, Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. Vol. I: xxxiv+464 pp. $ 15.00; Vol. II, 1965. MR**0174165****[2]**Earl A. Coddington,*An introduction to ordinary differential equations*, Prentice-Hall Mathematics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR**0126573****[3]**Jim Douglas Jr. and Todd Dupont,*A finite element collocation method for quasilinear parabolic equations*, Math. Comp.**27**(1973), 17–28. MR**0339508**, 10.1090/S0025-5718-1973-0339508-8**[4]**Jim Douglas Jr. and Todd Dupont,*A finite element collocation method for the heat equation*, Symposia Mathematica, Vol. X (Convegno di Analisi Numerica, INDAM, Rome, 1972) Academic Press, London, 1972, pp. 403–410. MR**0373329****[5]**Jim Douglas Jr. and Todd Dupont,*Collocation methods for parabolic equations in a single space variable*, Lecture Notes in Mathematics, Vol. 385, Springer-Verlag, Berlin-New York, 1974. Based on 𝐶¹-piecewise-polynomial spaces. MR**0483559****[6]**E. N. HOUSTIS,*Finite Element Methods for Solving Initial/Boundary Value Problems*, Doctoral thesis, Purdue University, 1974.**[7]**L. V. KANTOROVIČ, "Sur une méthode de resolution approchée d'equations différentielles aux derivées partielles,"*C. R. Acad.*(*Dokl.*)*Sci. URSS*, v. 2, 1934, pp. 532-536. (Russian)**[8]**È. B. Karpilovskaja,*Convergence of a collocation method for certain boundary-value problems of mathematical physics*, Sibirsk. Mat. Ž.**4**(1963), 632–640 (Russian). MR**0156479****[9]**Martin H. Schultz,*Spline analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR**0362832****[10]**Ju. P. Jarcev,*The convergence of the method of collocation by lines*, Differencial′nye Uravnenija**3**(1967), 1606–1613 (Russian). MR**0221768****[11]**Yu. P. YARTSEV, "The method of line collocation,"*Differencial'nye Uravnenija*, v. 4, 1968, pp. 925-932 =*Differential Equations*, v. 4, 1968, pp. 481-485.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1977-0443379-2

Keywords:
Collocation on lines method,
nonlinear hyperbolic problems

Article copyright:
© Copyright 1977
American Mathematical Society