Nothing Special   »   [go: up one dir, main page]
Skip to main content
SpringerLink
Log in

Automatic grid control in adaptive BVP solvers

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Grid adaptation in two-point boundary value problems is usually based on mapping a uniform auxiliary grid to the desired nonuniform grid. Here we combine this approach with a new control system for constructing a grid density function ϕ(x). The local mesh width Δx j + 1/2 = x j + 1 − x j with 0 = x 0 < x 1 < ... < x N  = 1 is computed as Δx j + 1/2 = ε N / φ j + 1/2, where \(\{\varphi_{j+1/2}\}_0^{N-1}\) is a discrete approximation to the continuous density function ϕ(x), representing mesh width variation. The parameter ε N  = 1/N controls accuracy via the choice of N. For any given grid, a solver provides an error estimate. Taking this as its input, the feedback control law then adjusts the grid, and the interaction continues until the error has been equidistributed. Digital filters may be employed to process the error estimate as well as the density to ensure the regularity of the grid. Once ϕ(x) is determined, another control law determines N based on the prescribed tolerance \({\textsc {tol}}\). The paper focuses on the interaction between control system and solver, and the controller’s ability to produce a near-optimal grid in a stable manner as well as correctly predict how many grid points are needed. Numerical tests demonstrate the advantages of the new control system within the bvpsuite solver, ceteris paribus, for a selection of problems and over a wide range of tolerances. The control system is modular and can be adapted to other solvers and error criteria.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ascher, U., Bader, U.: A new basis implementation for a mixed order boundary value ODE solver. SIAM Sci. J. Stat. Comput. 8, 483–500 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ascher, U., Christiansen, J., Russell, R.D.: A collocation solver for mixed order systems of boundary value problems. Math. Comput. 33, 659–679 (1978)

    MathSciNet  Google Scholar 

  3. Ascher, U., Christiansen, J., Russell, R.D.: Collocation Software for boundary-value ODEs. ACM Trans. Math. Softw. 7, 209–222 (1981)

    Article  MATH  Google Scholar 

  4. Ascher, U., Mattheij, R.M.M., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs (1988)

    MATH  Google Scholar 

  5. Auzinger, W., Koch, O., Weinmüller, E.B.: Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31, 5–25 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Auzinger, W., Kneisl, G., Koch, O., Weinmüller, E.B.: A collocation code for boundary value problems in ordinary differential equations. Numer. Algorithms 33, 27–39 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Auzinger, W., Koch, O., Pulverer, G., Weinmüller, E.B.: Performance of collocation software for singular BVPs. Technical Report, ANUM Preprint No. 4/04, Vienna University of Technology, Austria (2004)

  8. Auzinger, W., Koch, O., Weinmüller, E.B.: Efficient mesh selection for collocation methods applied to singular BVPs. J. Comput. Appl. Math. 180, 213–227 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Babuska, I., Rheinboldt, W.C.: Analysis of optimal finite-elemt meshes in R1, Math. Comput. 33, 435–463 (1979)

    MATH  MathSciNet  Google Scholar 

  10. Beckett, G., Mackenzie, J.A., Ramage, A., Slone, D.M.: On the numerical solution of one-dimensional pdes using adaptive methods based on equidistribution. Comput. J. Phys. 167, 372–392 (2001)

    Article  MATH  Google Scholar 

  11. de Boor, C.: Good approximations by splines with variable knots. In: Meir, A., Sharma, A. (eds.) Spline Functions and Approximation Theory, pp. 57–73. Birkhüser, Basel (1973)

  12. Budd, C.J., Koch, O., Weinmüller, E.B.: Self-similar blow-up in nonlinear PDEs. AURORA Technical Report, TR-2004-15, Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria (2004). Available at http://www.vcpc.univie.ac.at/aurora/publications/.

  13. Budd, C.J., Koch, O., Weinmüller, E.B.: Computation of self-similar solution profiles for the nonlinear Schrödinger equation. Computing 77, 335–346 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Budd, C.J., Koch, O., Weinmüller, E.B.: From nonlinear PDEs to singular ODEs. Appl. Numer. Math. 56, 413–422 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Carey, G.F., Dinh, H.T.: Grading functions and mesh redistribution. SIAM J. Numer. Anal. 22, 1028–1040 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chen, K.: Error equidistribution and mesh adaptation. SIAM J. Sci. Comput. 15, 798–818 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Christara, C.C., Ng, K.S.: Optimal quadratic and cubic spline collocation on nonuniform partitions. Computing 76, 227–257 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Christara, C.C., Ng, K.S.: Adaptive techniques for spline collocation. Computing 76, 259–277 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. De Hoog, F., Weiss, R.: Difference methods for boundary value problems with a singularity of the first kind. SIAM J. Numer. Anal. 13, 775–813 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Huang, W., Ren, Y., Russell, R.D.: Moving mesh partial differential equations (MMPDEs) based on the equidistribution principle. SIAM J. Numer. Anal. 31, 709 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Huang, W.: Variational mesh adaptation: isotropy and equidistribution. J. Comput. Phys. 174, 903–924 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Huang, W., Sun, W.: Variational mesh adaptation II: error estimates and monitor functions. SIAM J. Numer. Anal. 13, 775–813 (1976)

    Article  MathSciNet  Google Scholar 

  23. Ilie, S., Söderlind, G., Corless, R.: Adaptivity and computational complexity in the numerical solution of ODEs. J. Complex. 24, 341–361 (2008)

    Article  MATH  Google Scholar 

  24. Kierzenka, J., Shampine, L.: A BVP solver that controls residual and error. JNAIAM Numer. J. Anal. Indust. Math. 3, 27–41 (2008)

    MATH  MathSciNet  Google Scholar 

  25. Kitzhofer, G.: Numerical treatment of implicit singular BVPs. Ph.D. thesis, Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria

  26. Kitzhofer, G., Koch, O., Weinmüller, E.B.: Collocation methods for the computation of bubble-type solutions of a singular boundary value problem in hydrodynamics. J. Sci. Comp. 32, 411–424 (2007). Available at http://www.math.tuwien.ac.at/~ewa

    Article  MATH  Google Scholar 

  27. Pereyra, V., Sewell, E.G.: Mesh selection for discrete solution of boundary problems in ordinary differential equations. Numer. Math. 23, 261–268 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  28. Polster, W.: Ein Algorithmus zur Gittersteuerung bei Kollokationsverfahren für singuläre Randwertprobleme. Master thesis, Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Austria (2001)

  29. Pryce, J.D.: On the convergence of iterated remeshing. IMA J. Numer. Anal. 9, 315–335 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rachunková, I., Koch, O., Pulverer, G., Weinmüller, E.B.: On a singular boundary value problem arising in the theory of shallow membrane caps. J. Math. Anal. Appl. 332, 523–541 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Rentrop, P.: Eine Taylorreihenmethode zur numerischen Lösung von Zwei-Punkt Randwertproblemen mit Anwendung auf singuläre Probleme der nichtlinearen Schalentheorie, TUM-MATH-7733, Technische Universität München (1977)

  32. Russell, R.D., Christiansen, J.: Adaptive mesh selection strategies for solving boundary value problems. SIAM J. Numer. Anal. 15, 59–80 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  33. Shampine, L., Kierzenka, J., Reichelt, M.: Solving boundary value problems for ordinary differential equations in Matlab with bvp4c (2000). Available at ftp://ftp.mathworks.com/pub/doc/papers/bvp/

  34. Shampine, L., Kierzenka, J.: A BVP solver based on residual control and the MATLAB PSE. ACM Trans. Math. Softw. 27, 299–315 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Shampine, L., Muir, P., Xu, H.: A user-friendly Fortran BVP solver. Available at http://cs.smu.ca/~muir/BVP_SOLVER_Files/ShampineMuirXu2006.pdf

  36. Söderlind, G.: Digital filters in adaptive time-stepping. ACM Trans. Math. Softw. 29, 1–26 (2003)

    Article  MATH  Google Scholar 

  37. Söderlind, G.: Time-step selection algorithms: adaptivity, control and signal processing. Appl. Numer. Math. 56, 488–502 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  38. Stockie, J.M., Mackenzie, J.A., Russell, R.D.: A moving mesh method for one-dimensional hyperbolic conservation laws. SISC 22, 1791–1813 (2001)

    MATH  MathSciNet  Google Scholar 

  39. Tang, H., Tang, T.: Adaptive methods for one- and two-dimensional hyperbolic conservation laws. SINUM 41, 487–515 (2003)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, 1040, Vienna, Austria

    Gernot Pulverer & Ewa Weinmüller

  2. Numerical Analysis, Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00, Lund, Sweden

    Gustaf Söderlind

Authors
  1. Gernot Pulverer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gustaf Söderlind
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ewa Weinmüller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gustaf Söderlind.

Additional information

Gustaf Söderlind was supported by Swedish Research Council grant VR 621-2005-3129.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pulverer, G., Söderlind, G. & Weinmüller, E. Automatic grid control in adaptive BVP solvers. Numer Algor 56, 61–92 (2011). https://doi.org/10.1007/s11075-010-9374-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-010-9374-0

Keywords

Search

Navigation