Deferred Corrections Software and Its Application to Seismic Ray Tracing

  • V. Pereyra
Part of the Computing Supplementum book series (COMPUTING, volume 5)


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.


Adaptive Mesh Refinement Defect Correction High Order Interpolation Global Error Estimation Deferred Correction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ascher, U., Christiansen, J., Russell, R. D.: A collocation solver for mixed order systems of BVP’s. Math. Comp. 33, 659–679 (1978).MathSciNetCrossRefGoogle Scholar
  2. [2]
    Ascher, U., Russell, R. D.: Reformulation of BVP’s into “standard”, form. SIAM Rev. 23, 238–254 (1981).MathSciNetMATHCrossRefGoogle Scholar
  3. [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. [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. [5]
    Björck, A., Pereyra, V.: Solution of Vandermonde systems of equations. Math. Comp. 24, 893–904 (1970).MathSciNetCrossRefGoogle Scholar
  6. [6]
    Bolstad, J.: PhD Thesis, Stanford Univ. California, U.S.A., 1982.Google Scholar
  7. [7]
    de Boor, C.: Good approximation by splines with variable knots. Lecture Notes in Mathematics 363, 12–20 (1973).CrossRefGoogle Scholar
  8. [8]
    Burchard, H. G.: Splines (with optimal knots) are better. J. Appl. Anal. 3, 309–319 (1974).MathSciNetMATHCrossRefGoogle Scholar
  9. [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. [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. [11]
    Christiansen, J., Russell, R. D.: Deferred corrections using non-symmetric end formulas. Numer. Math. 35, 21–33 (1980).MathSciNetMATHCrossRefGoogle Scholar
  12. [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).MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Daniel, J. W., Pereyra, V., Schumaker, L. L.: Iterated deferred corrections for IVP’s. Acta Cient. Venezolana 19, 128–135 (1968).MathSciNetGoogle Scholar
  14. [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).CrossRefGoogle Scholar
  15. [15]
    Hackbusch, W., Trottenberg, U. (eds.): Multigrid Methods. (Lecture Notes in Mathematics, Vol. 960.) Berlin-Heidelberg-New York: Springer 1982.MATHGoogle Scholar
  16. [16]
    Henrici, P.: Discrete Variable Methods in ODE’s. New York: Wiley 1962.Google Scholar
  17. [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. [18]
    Keller, H. B.: Numerical Methods for 2PBVP’s. Waltham, Mass.: Blaisdell 1968.Google Scholar
  19. [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. [20]
    Keller, H. B., Pereyra, V.: Difference methods and deferred corrections for OBVP’s. SIAM J. Numer. Anal. 16, 241–259 (1979).MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Keller, H. B., White, A. B.: Difference methods for BVP’s in ODE’s. SIAM J. Numer. Anal. 12, 791–801 (1975).MathSciNetMATHCrossRefGoogle Scholar
  22. [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. [23]
    Lentini, M., Pereyra, V.: A variable order finite difference method for nonlinear multipoint BVP’s. Math. Comp. 28, 981–1003 (1974).MathSciNetMATHCrossRefGoogle Scholar
  24. [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. [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).MathSciNetMATHCrossRefGoogle Scholar
  26. [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).MathSciNetMATHGoogle Scholar
  27. [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. [28]
    Pearson, C. E.: On a differential equation of boundary layer type. J. Math. Phys. 47, 134–154 (1968).MATHGoogle Scholar
  29. [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. [30]
    Pereyra, V.: The correction difference method for non-linear BVP’s of class. M. Rev. Unión Matemätica Argentina 22, 184–201 (1965).MathSciNetMATHGoogle Scholar
  31. [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. [32]
    Pereyra, V.: Accelerating the convergence of discretization algorithms. SIAM J. Numer. Anal. 4, 508–533 (1967).MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Pereyra, V.: Iterated deferred corrections for nonlinear operator equations. Numer. Math. 10, 316–323 (1967).MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Pereyra, V.: Iterated deferred corrections for nonlinear BVP’s. Numer. Math. 11, 111–125 (1968).MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    Pereyra, V.: Highly accurate numerical solution of casilinear elliptic BVP’s in n dimensions. Math. Comp. 24, 771–783 (1970).MathSciNetMATHGoogle Scholar
  36. [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. [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. [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).MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    Pereyra, V.: An adaptive finite difference FORTRAN program for first order. NLOBP’s. In [9], pp. 67–88 (1979).Google Scholar
  40. [40]
    Pereyra, V.: Solución numérica de ecuaciones diferenciales con condiciones de frontera. Acta Cient. Venezolana 30, 7–22 (1979).MathSciNetMATHGoogle Scholar
  41. [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. [42]
    Pereyra, V.: Constrained ill-conditioned non-linear least squares. (In preparation.)Google Scholar
  43. [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).CrossRefGoogle Scholar
  44. [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. [45]
    Pereyra, V., Sewell, G.: Mesh selection for discrete solution of BVP’s in ODE’s. Numer. Math. 23, 261–268 (1975).MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    Pereyra, V., Wojcik, G.: Interactive seismic ray tracing on complex geological structures. (In preparation.)Google Scholar
  47. [47]
    Russell, R. D.: Mesh selection methods. In [9], pp. 228–242 (1979).Google Scholar
  48. [48]
    Russell, R. D., Christiansen, J.: Adaptive mesh selection strategies for solving BVP’s. SIAM J. Numer. Anal. 15, 59–80 (1978).MathSciNetMATHCrossRefGoogle Scholar
  49. [49]
    Sewell, E. G.: Automatic generation of triangulations for piecewise polynomial approximation. PhD Thesis, Purdue Univ. Indiana, U.S.A., 1972.Google Scholar
  50. [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. [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. [52]
    Simpson, R. B.: Automatical local refinements for irregular meshes. Research Rep. CS-78-19. Univ. Waterloo, Ontario, Canada, 1978.Google Scholar
  53. [53]
    Stetter, H. J.: Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations. Numer. Math. 7, 18–31 (1965).MathSciNetMATHCrossRefGoogle Scholar
  54. [54]
    Stetter, H. J.: Global error estimation in Adams PC.-codes. ACM Trans. Math. Software 5, 415–430 (1979).MATHCrossRefGoogle Scholar
  55. [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).MathSciNetMATHCrossRefGoogle Scholar
  56. [56]
    White, A. B.: On selection of equidistributing meshes for TPBVP’s. SIAM J. Numer. Anal. 16, 472–502 (1979).MathSciNetMATHCrossRefGoogle Scholar
  57. [57]
    Wilts, C.: Ferromagnetic resonance equations for implanted films. Manuscript. Dept. Electrical Eng., Caltech. Pasadena, Ca., U.S.A., 1980.Google Scholar
  58. [58]
    Zadunaisky, P. E.: On the estimation of errors propagated in the numerical integration of ODE’s. Numer. Math. 27, 21–40 (1976).MathSciNetMATHCrossRefGoogle Scholar
  59. [59]
    Keller, H. B.: Accurate difference methods for NLTPBVP’s. SIAM J. Numer. Anal. 11, 305–320 (1974).MathSciNetMATHCrossRefGoogle Scholar
  60. [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. [61]
    Varah, J. M.: On the solution of block tridiagonal systems arising from certain finite-difference equations. Math. Comp. 26, 859–868 (1972).MathSciNetMATHCrossRefGoogle Scholar
  62. [62]
    Varah, J. M.: Alternate row and column elimination for solving certain linear systems. SIAM J. Numer. Anal. 13, 71–75 (1976).MathSciNetMATHCrossRefGoogle Scholar
  63. [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. [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. [65]
    Böhmer, K.: Discrete Newton methods and iterated defect corrections. Numer. Math. 37, 167–192 (1981).MathSciNetMATHCrossRefGoogle Scholar
  66. [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.CrossRefGoogle Scholar
  67. [67]
    Lindberg, B.: Error estimation and iterative improvement for discretization algorithms. BIT 20, 486–500 (1980).MathSciNetMATHCrossRefGoogle Scholar
  68. [68]
    Frank, R., Hertling, J., Ueberhuber, C.: An extension of the applicability of iterated deferred corrections. Math. Comp. 31, 907–915 (1977).MathSciNetMATHGoogle Scholar
  69. [69]
    Stetter, H. J.: The defect correction principle and discretization methods. Num. Math. 29, 425–443 (1978).MathSciNetMATHCrossRefGoogle Scholar
  70. [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. [71]
    Skeel, R. D.: A theoretical framework for proving accuracy results for deferred corrections. SIAM J. Numer. Anal. 19, 171–196 (1982).MathSciNetMATHCrossRefGoogle Scholar
  72. [72]
    Kreiss, H.-O., Nichols, N.: Numerical methods for singular perturbation problems. Rep. 57, Dept. Comp. Sc., Uppsala Univ., Sweden, 1975.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • V. Pereyra
    • 1
  1. 1.Computer Sc. Dept.Stanford UniversityStanfordUSA

Personalised recommendations