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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ascher, U., Christiansen, J., Russell, R. D.: A collocation solver for mixed order systems of BVP’s. Math. Comp. 33, 659–679 (1978).
Ascher, U., Russell, R. D.: Reformulation of BVP’s into “standard”, form. SIAM Rev. 23, 238–254 (1981).
Berger, M. J.: Adaptive mesh refinements for hyperbolic PDE’s. STAN-CS 82-924. Comp. Sc. Dept. Stanford Univ., California, U.S.A., 1982.
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.
Björck, A., Pereyra, V.: Solution of Vandermonde systems of equations. Math. Comp. 24, 893–904 (1970).
Bolstad, J.: PhD Thesis, Stanford Univ. California, U.S.A., 1982.
de Boor, C.: Good approximation by splines with variable knots. Lecture Notes in Mathematics 363, 12–20 (1973).
Burchard, H. G.: Splines (with optimal knots) are better. J. Appl. Anal. 3, 309–319 (1974).
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.
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.
Christiansen, J., Russell, R. D.: Deferred corrections using non-symmetric end formulas. Numer. Math. 35, 21–33 (1980).
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).
Daniel, J. W., Pereyra, V., Schumaker, L. L.: Iterated deferred corrections for IVP’s. Acta Cient. Venezolana 19, 128–135 (1968).
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).
Hackbusch, W., Trottenberg, U. (eds.): Multigrid Methods. (Lecture Notes in Mathematics, Vol. 960.) Berlin-Heidelberg-New York: Springer 1982.
Henrici, P.: Discrete Variable Methods in ODE’s. New York: Wiley 1962.
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).
Keller, H. B.: Numerical Methods for 2PBVP’s. Waltham, Mass.: Blaisdell 1968.
Keller, H. B.: Numerical Solution of TPBVP’s. Reg. Conf. Series Appl. Math. 24, SIAM, Philadelphia, Pa., U.S.A., 1976.
Keller, H. B., Pereyra, V.: Difference methods and deferred corrections for OBVP’s. SIAM J. Numer. Anal. 16, 241–259 (1979).
Keller, H. B., White, A. B.: Difference methods for BVP’s in ODE’s. SIAM J. Numer. Anal. 12, 791–801 (1975).
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.
Lentini, M., Pereyra, V.: A variable order finite difference method for nonlinear multipoint BVP’s. Math. Comp. 28, 981–1003 (1974).
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.
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).
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).
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.
Pearson, C. E.: On a differential equation of boundary layer type. J. Math. Phys. 47, 134–154 (1968).
Pereyra, V.: The difference correction method for non-linear TPBVP’s. Techn. Rep. CS18, Comp. Sc. Dept., Stanford Univ. California, U.S.A., 1965.
Pereyra, V.: The correction difference method for non-linear BVP’s of class. M. Rev. Unión Matemätica Argentina 22, 184–201 (1965).
Pereyra, V.: Highly accurate discrete methods for nonlinear problems. PhD Thesis, Comp. Sc. Dept., Univ. Wisconsin. Madison, Wis., U.S.A., 1967.
Pereyra, V.: Accelerating the convergence of discretization algorithms. SIAM J. Numer. Anal. 4, 508–533 (1967).
Pereyra, V.: Iterated deferred corrections for nonlinear operator equations. Numer. Math. 10, 316–323 (1967).
Pereyra, V.: Iterated deferred corrections for nonlinear BVP’s. Numer. Math. 11, 111–125 (1968).
Pereyra, V.: Highly accurate numerical solution of casilinear elliptic BVP’s in n dimensions. Math. Comp. 24, 771–783 (1970).
Pereyra, V.: High order finite difference solution of differential equations. STAN-CS-73-348, Comp. Sc. Dept. Stanford Univ., California, U.S.A., 1973.
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.
Pereyra, V., Proskurowski, W., Widlund, O.: High order fast Laplace solvers for the Dirichlet problem on general regions. Math. Comp. 31, 1–16 (1977).
Pereyra, V.: An adaptive finite difference FORTRAN program for first order. NLOBP’s. In [9], pp. 67–88 (1979).
Pereyra, V.: Solución numérica de ecuaciones diferenciales con condiciones de frontera. Acta Cient. Venezolana 30, 7–22 (1979).
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.
Pereyra, V.: Constrained ill-conditioned non-linear least squares. (In preparation.)
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).
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).
Pereyra, V., Sewell, G.: Mesh selection for discrete solution of BVP’s in ODE’s. Numer. Math. 23, 261–268 (1975).
Pereyra, V., Wojcik, G.: Interactive seismic ray tracing on complex geological structures. (In preparation.)
Russell, R. D.: Mesh selection methods. In [9], pp. 228–242 (1979).
Russell, R. D., Christiansen, J.: Adaptive mesh selection strategies for solving BVP’s. SIAM J. Numer. Anal. 15, 59–80 (1978).
Sewell, E. G.: Automatic generation of triangulations for piecewise polynomial approximation. PhD Thesis, Purdue Univ. Indiana, U.S.A., 1972.
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.
Sewell, E. G.: TWODEPEP: A small general purpose finite element program. Techn. Rep. 8102, IMSL. Houston Texas, U.S.A., 1981.
Simpson, R. B.: Automatical local refinements for irregular meshes. Research Rep. CS-78-19. Univ. Waterloo, Ontario, Canada, 1978.
Stetter, H. J.: Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations. Numer. Math. 7, 18–31 (1965).
Stetter, H. J.: Global error estimation in Adams PC.-codes. ACM Trans. Math. Software 5, 415–430 (1979).
White, A. B.: On the numerical solution of initial/boundary value problems in one space dimension. SIAM J. Numer. Anal. 19, 683–697 (1982).
White, A. B.: On selection of equidistributing meshes for TPBVP’s. SIAM J. Numer. Anal. 16, 472–502 (1979).
Wilts, C.: Ferromagnetic resonance equations for implanted films. Manuscript. Dept. Electrical Eng., Caltech. Pasadena, Ca., U.S.A., 1980.
Zadunaisky, P. E.: On the estimation of errors propagated in the numerical integration of ODE’s. Numer. Math. 27, 21–40 (1976).
Keller, H. B.: Accurate difference methods for NLTPBVP’s. SIAM J. Numer. Anal. 11, 305–320 (1974).
Lam, D. C. L.: Implementation of the Box scheme and model analysis of diffusion-convection equations. PhD Thesis, Univ. Waterloo. Ontario, Canada (1974).
Varah, J. M.: On the solution of block tridiagonal systems arising from certain finite-difference equations. Math. Comp. 26, 859–868 (1972).
Varah, J. M.: Alternate row and column elimination for solving certain linear systems. SIAM J. Numer. Anal. 13, 71–75 (1976).
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.
Lentini, M., Osborne, M. R., Russell, R. D.: The close relationship between methods for solving TPBVP’s. SIAM J. Numer. Anal. (Submitted for publication.)
Böhmer, K.: Discrete Newton methods and iterated defect corrections. Numer. Math. 37, 167–192 (1981).
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.
Lindberg, B.: Error estimation and iterative improvement for discretization algorithms. BIT 20, 486–500 (1980).
Frank, R., Hertling, J., Ueberhuber, C.: An extension of the applicability of iterated deferred corrections. Math. Comp. 31, 907–915 (1977).
Stetter, H. J.: The defect correction principle and discretization methods. Num. Math. 29, 425–443 (1978).
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.
Skeel, R. D.: A theoretical framework for proving accuracy results for deferred corrections. SIAM J. Numer. Anal. 19, 171–196 (1982).
Kreiss, H.-O., Nichols, N.: Numerical methods for singular perturbation problems. Rep. 57, Dept. Comp. Sc., Uppsala Univ., Sweden, 1975.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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