Skip to main content

Deferred Corrections Software and Its Application to Seismic Ray Tracing

  • Chapter
Defect Correction Methods

Part of the book series: Computing Supplementum ((COMPUTING,volume 5))

Abstract

We give first a historical account of the various stages of development of iterated deferred corrections software, mainly for ordinary two-point boundary value problems, but mentioning also some work on partial differential equations. Then we describe the latest code on the PASVA series (No. 4), which extends the earlier one to problems with discontinuous data and mixed systems of differential and algebraic conditions. Finally, an example of application of this code to two-point ray tracing on piece-wise continuous media is given.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ascher, U., Christiansen, J., Russell, R. D.: A collocation solver for mixed order systems of BVP’s. Math. Comp. 33, 659–679 (1978).

    Article  MathSciNet  Google Scholar 

  2. Ascher, U., Russell, R. D.: Reformulation of BVP’s into “standard”, form. SIAM Rev. 23, 238–254 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  3. Berger, M. J.: Adaptive mesh refinements for hyperbolic PDE’s. STAN-CS 82-924. Comp. Sc. Dept. Stanford Univ., California, U.S.A., 1982.

    Google Scholar 

  4. Berger, M. J., Oliger, J.: Adaptive mesh refinement for hyperbolic PDE’s. NA-83-02, Comp. Sc. Dept. Stanford Univ., California, U.S.A., 1983.

    Google Scholar 

  5. Björck, A., Pereyra, V.: Solution of Vandermonde systems of equations. Math. Comp. 24, 893–904 (1970).

    Article  MathSciNet  Google Scholar 

  6. Bolstad, J.: PhD Thesis, Stanford Univ. California, U.S.A., 1982.

    Google Scholar 

  7. de Boor, C.: Good approximation by splines with variable knots. Lecture Notes in Mathematics 363, 12–20 (1973).

    Article  Google Scholar 

  8. Burchard, H. G.: Splines (with optimal knots) are better. J. Appl. Anal. 3, 309–319 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  9. Childs, B., Scott, M., Daniel, J. W., Denman, E., Nelson, P. (eds.): Codes for BVP’s in ODE’s. (Lecture Notes Comp. Sc., Vol. 76.) Berlin-Heidelberg-New York: Springer 1979.

    Google Scholar 

  10. Corless, R. M.: A feasibility study of the use of COLSYS in the solution of systems of PDE’s. CS-82-32, Dept. Appl. Math. Univ. Waterloo, Ontario, Canada, 1982.

    Google Scholar 

  11. Christiansen, J., Russell, R. D.: Deferred corrections using non-symmetric end formulas. Numer. Math. 35, 21–33 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  12. Daniel, J. W., Martin, A. J.: Numerov’s method with deferred corrections for two-point boundary-value problems. SIAM J. Numer. Anal. 14, 1033–1050 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  13. Daniel, J. W., Pereyra, V., Schumaker, L. L.: Iterated deferred corrections for IVP’s. Acta Cient. Venezolana 19, 128–135 (1968).

    MathSciNet  Google Scholar 

  14. Fox, L.: Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations. Proc. Royal Soc. (London) A 190, 31–59 (1947).

    Article  Google Scholar 

  15. Hackbusch, W., Trottenberg, U. (eds.): Multigrid Methods. (Lecture Notes in Mathematics, Vol. 960.) Berlin-Heidelberg-New York: Springer 1982.

    MATH  Google Scholar 

  16. Henrici, P.: Discrete Variable Methods in ODE’s. New York: Wiley 1962.

    Google Scholar 

  17. Itoh, K.: On the numerical solvability of 2 PBVP’s in a finite Chebyshev series for piece-wise smooth differential systems. Mem. Numer. Math. 2, 45–67 (1975).

    Google Scholar 

  18. Keller, H. B.: Numerical Methods for 2PBVP’s. Waltham, Mass.: Blaisdell 1968.

    Google Scholar 

  19. Keller, H. B.: Numerical Solution of TPBVP’s. Reg. Conf. Series Appl. Math. 24, SIAM, Philadelphia, Pa., U.S.A., 1976.

    Google Scholar 

  20. Keller, H. B., Pereyra, V.: Difference methods and deferred corrections for OBVP’s. SIAM J. Numer. Anal. 16, 241–259 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  21. Keller, H. B., White, A. B.: Difference methods for BVP’s in ODE’s. SIAM J. Numer. Anal. 12, 791–801 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  22. Lentini, M.: Correcciones diferidas para problemas de contorno en sistemas de ecuaciones diferenciales ordinarias de primer orden. Pub. 73-04, Dpto. Comp., Fac. Ciencias. Univ. Central, Caracas, Venezuela, 1973.

    Google Scholar 

  23. Lentini, M., Pereyra, V.: A variable order finite difference method for nonlinear multipoint BVP’s. Math. Comp. 28, 981–1003 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  24. Lentini, M., Pereyra, V.: Boundary problem solvers for first order systems based on deferred corrections. In: Numerical Solution of BVP’s for ODE’s (Aziz, A. K., ed.). New York: Academic Press 1975.

    Google Scholar 

  25. Lentini, M., Pereyra, V.: An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers. SIAM J. Numer. Anal. 14, 91–111 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  26. Lentini, M., Pereyra, V.: PASVA4: An ordinary boundary solver for problems with discontinuous interfaces and algebraic parameters. Mat. Aplicada e Comp. 2, 103–118 (1983).

    MathSciNet  MATH  Google Scholar 

  27. Lopez, H., Ruiz, L.: Extrapolaciones sucesivas para problemas de contorno en sistemas no-lineales de EDO’s. Trabajo Esp. de Grado. Depto. Comp., Fac. Ciencias. Univ. Central, Caracas, Venezuela, 1974.

    Google Scholar 

  28. Pearson, C. E.: On a differential equation of boundary layer type. J. Math. Phys. 47, 134–154 (1968).

    MATH  Google Scholar 

  29. Pereyra, V.: The difference correction method for non-linear TPBVP’s. Techn. Rep. CS18, Comp. Sc. Dept., Stanford Univ. California, U.S.A., 1965.

    Google Scholar 

  30. Pereyra, V.: The correction difference method for non-linear BVP’s of class. M. Rev. Unión Matemätica Argentina 22, 184–201 (1965).

    MathSciNet  MATH  Google Scholar 

  31. Pereyra, V.: Highly accurate discrete methods for nonlinear problems. PhD Thesis, Comp. Sc. Dept., Univ. Wisconsin. Madison, Wis., U.S.A., 1967.

    Google Scholar 

  32. Pereyra, V.: Accelerating the convergence of discretization algorithms. SIAM J. Numer. Anal. 4, 508–533 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  33. Pereyra, V.: Iterated deferred corrections for nonlinear operator equations. Numer. Math. 10, 316–323 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  34. Pereyra, V.: Iterated deferred corrections for nonlinear BVP’s. Numer. Math. 11, 111–125 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  35. Pereyra, V.: Highly accurate numerical solution of casilinear elliptic BVP’s in n dimensions. Math. Comp. 24, 771–783 (1970).

    MathSciNet  MATH  Google Scholar 

  36. Pereyra, V.: High order finite difference solution of differential equations. STAN-CS-73-348, Comp. Sc. Dept. Stanford Univ., California, U.S.A., 1973.

    Google Scholar 

  37. Pereyra, V.: Variable order variable step finite difference methods for NLBVP’s. (Lecture Notes in Mathematics, Vol. 363, pp. 118–133.) Berlin-Heidelberg-New York: Springer 1974.

    Google Scholar 

  38. Pereyra, V., Proskurowski, W., Widlund, O.: High order fast Laplace solvers for the Dirichlet problem on general regions. Math. Comp. 31, 1–16 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  39. Pereyra, V.: An adaptive finite difference FORTRAN program for first order. NLOBP’s. In [9], pp. 67–88 (1979).

    Google Scholar 

  40. Pereyra, V.: Solución numérica de ecuaciones diferenciales con condiciones de frontera. Acta Cient. Venezolana 30, 7–22 (1979).

    MathSciNet  MATH  Google Scholar 

  41. Pereyra, V.: Two-point ray tracing in heterogeneous media and the inversion of travel time data. In: Computing Methods in Appl. Sc. and Eng. (Glowinski, R., Lions, J. L., eds.), pp. 553–570. Amsterdam: North-Holland 1980.

    Google Scholar 

  42. Pereyra, V.: Constrained ill-conditioned non-linear least squares. (In preparation.)

    Google Scholar 

  43. Pereyra, V., Keller, H. B., Lee, W. H. K.: Computational methods for inverse problems in geophysics: inversion of travel time observations. Physics of the Earth and Planetary Interiors 21, 120–125 (1980).

    Article  Google Scholar 

  44. Pereyra, V., Lee, W. H. K., Keller, H. B.: Solving two-point ray tracing problems in a heterogeneous medium. Bull. Seism. Soc. America 70, 79–99 (1980).

    Google Scholar 

  45. Pereyra, V., Sewell, G.: Mesh selection for discrete solution of BVP’s in ODE’s. Numer. Math. 23, 261–268 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  46. Pereyra, V., Wojcik, G.: Interactive seismic ray tracing on complex geological structures. (In preparation.)

    Google Scholar 

  47. Russell, R. D.: Mesh selection methods. In [9], pp. 228–242 (1979).

    Google Scholar 

  48. Russell, R. D., Christiansen, J.: Adaptive mesh selection strategies for solving BVP’s. SIAM J. Numer. Anal. 15, 59–80 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  49. Sewell, E. G.: Automatic generation of triangulations for piecewise polynomial approximation. PhD Thesis, Purdue Univ. Indiana, U.S.A., 1972.

    Google Scholar 

  50. Sewell, E. G.: An adaptive computer program for the solution of Div (P(x, y) Grad u) = f(x, y, u) on a polygonal region. In: MAFELAP II (Whiteman, J. R., ed.), pp. 543–553. New York: Academic Press 1976.

    Google Scholar 

  51. Sewell, E. G.: TWODEPEP: A small general purpose finite element program. Techn. Rep. 8102, IMSL. Houston Texas, U.S.A., 1981.

    Google Scholar 

  52. Simpson, R. B.: Automatical local refinements for irregular meshes. Research Rep. CS-78-19. Univ. Waterloo, Ontario, Canada, 1978.

    Google Scholar 

  53. Stetter, H. J.: Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations. Numer. Math. 7, 18–31 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  54. Stetter, H. J.: Global error estimation in Adams PC.-codes. ACM Trans. Math. Software 5, 415–430 (1979).

    Article  MATH  Google Scholar 

  55. White, A. B.: On the numerical solution of initial/boundary value problems in one space dimension. SIAM J. Numer. Anal. 19, 683–697 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  56. White, A. B.: On selection of equidistributing meshes for TPBVP’s. SIAM J. Numer. Anal. 16, 472–502 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  57. Wilts, C.: Ferromagnetic resonance equations for implanted films. Manuscript. Dept. Electrical Eng., Caltech. Pasadena, Ca., U.S.A., 1980.

    Google Scholar 

  58. Zadunaisky, P. E.: On the estimation of errors propagated in the numerical integration of ODE’s. Numer. Math. 27, 21–40 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  59. Keller, H. B.: Accurate difference methods for NLTPBVP’s. SIAM J. Numer. Anal. 11, 305–320 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  60. Lam, D. C. L.: Implementation of the Box scheme and model analysis of diffusion-convection equations. PhD Thesis, Univ. Waterloo. Ontario, Canada (1974).

    Google Scholar 

  61. Varah, J. M.: On the solution of block tridiagonal systems arising from certain finite-difference equations. Math. Comp. 26, 859–868 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  62. Varah, J. M.: Alternate row and column elimination for solving certain linear systems. SIAM J. Numer. Anal. 13, 71–75 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  63. Markowich, P. A., Ringhofer, Chr. A., Selberherr, S., Langer, E.: A singularly perturbed BVP modelling a semiconductor device. Techn. Summ. Rep. No. 2388. Math. Res. Center, Univ. Wisconsin, Madison, U.S.A., 1982.

    Google Scholar 

  64. Lentini, M., Osborne, M. R., Russell, R. D.: The close relationship between methods for solving TPBVP’s. SIAM J. Numer. Anal. (Submitted for publication.)

    Google Scholar 

  65. Böhmer, K.: Discrete Newton methods and iterated defect corrections. Numer. Math. 37, 167–192 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  66. Allgöwer, E. L., Böhmer, K., Mc Cormick, S.: Discrete correction methods for operator equations. In: Numerical Solution of Nonlinear Equations (Lecture Notes in Mathematics, Vol. 878), pp. 31–97. Berlin-Heidelberg-New York: Springer 1981.

    Chapter  Google Scholar 

  67. Lindberg, B.: Error estimation and iterative improvement for discretization algorithms. BIT 20, 486–500 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  68. Frank, R., Hertling, J., Ueberhuber, C.: An extension of the applicability of iterated deferred corrections. Math. Comp. 31, 907–915 (1977).

    MathSciNet  MATH  Google Scholar 

  69. Stetter, H. J.: The defect correction principle and discretization methods. Num. Math. 29, 425–443 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  70. Zadunaisky, P. E.: A method for the estimation of errors propagated in the numerical solution of a system of ODE’s. Proc. Astron. Union Symp., No. 25. New York: Academic Press 1966.

    Google Scholar 

  71. Skeel, R. D.: A theoretical framework for proving accuracy results for deferred corrections. SIAM J. Numer. Anal. 19, 171–196 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  72. Kreiss, H.-O., Nichols, N.: Numerical methods for singular perturbation problems. Rep. 57, Dept. Comp. Sc., Uppsala Univ., Sweden, 1975.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag

About this chapter

Cite this chapter

Pereyra, V. (1984). Deferred Corrections Software and Its Application to Seismic Ray Tracing. In: Böhmer, K., Stetter, H.J. (eds) Defect Correction Methods. Computing Supplementum, vol 5. Springer, Vienna. https://doi.org/10.1007/978-3-7091-7023-6_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-7023-6_12

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-81832-9

  • Online ISBN: 978-3-7091-7023-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics