Numerische Mathematik

, Volume 98, Issue 4, pp 581–606 | Cite as

Solution of lambda-omega systems: Theta-schemes and multigrid methods



The numerical solution of a time-dependent reaction diffusion lambda-omega system discretized by theta-schemes and finite differences is considered. Stability and accuracy of finite difference theta-schemes for this system are established. To solve the time-implicit evolution equations a nonlinear multigrid method is applied. The convergence properties of this solver are investigated considering a linearized lambda-omega model.


Finite Difference Evolution Equation Convergence Property Reaction Diffusion Multigrid Method 
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.



I would like to thank the anonymous Referees for many comments and suggestions.


  1. 1.
    Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)MATHGoogle Scholar
  2. 2.
    Benzi, M., Bertaccini, D.: Approximate inverse preconditioning for shifted linear systems. BIT Numer. Math. 43, 231–244 (2003)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Borzì, A.: Multigrid methods for parabolic distributed optimal control problems. J. Comp. Appl. Math 157, 365–382 (2003)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Borzì, A., Kunisch, K., Kwak Do, Y.: Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Optim. 41, 1477–1497 (2003)CrossRefGoogle Scholar
  5. 5.
    Bramble, J.H.: Multigrid Methods. Pitman Research Notes in Mathematics Series, Essex, 1993Google Scholar
  6. 6.
    Bramble, J.H., Kwak Do, Y., Pasciak, J.E.: Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems. SIAM J. Numer. Anal. 31, 1746–1763 (1994)MathSciNetMATHGoogle Scholar
  7. 7.
    Bramble, J.H., Pasciak, J.E., Sammon, P.H., Thomèe, V.: Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data. Math. Comput. 52, 339–367 (1989)MathSciNetMATHGoogle Scholar
  8. 8.
    Bramble, J.H., Pasciak, J.E., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comput. 56, 1–34 (1991)MathSciNetMATHGoogle Scholar
  9. 9.
    Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977)MATHGoogle Scholar
  10. 10.
    Brandt, A., Greenwald, J.: Parabolic multigrid revisited. In: W. Hackbusch, U. Trottenberg (eds.), Multigrid Methods III, Int. Series of Numerical Mathematics, Vol. 98, Birkhäuser, Basel, 1991Google Scholar
  11. 11.
    Browning, G.L., Kreiss, H.-O.: Comparison of numerical methods for the calculation of two-dimensional turbulence. Math. Comput. 52, 369–388 (1989)MathSciNetMATHGoogle Scholar
  12. 12.
    Chou, S.H., Kwak Do, Y.: V-Cycle multigrid for a vertex-centered covolume method for elliptic problems. Numer. Math. 90(3), 441–458 (2002)CrossRefMATHGoogle Scholar
  13. 13.
    Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)CrossRefGoogle Scholar
  14. 14.
    Douglas, J., Dupont, T., Jr.: Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 7(4), 575–626 (1970)MATHGoogle Scholar
  15. 15.
    Duffy, A., Britton, K., Murray, J.: Spiral wave solutions of practical reaction-diffusion systems. SIAM J. Appl. Math 39(1), 8–13 (1980)MATHGoogle Scholar
  16. 16.
    Eguiluz, V.M., Hernandez-Garcia, E., Piro, O.: Complex Ginzburg-Landau equation in the presence of walls and corners. Phys. Rev. E 64, 1–10 (2001)Google Scholar
  17. 17.
    Hackbusch, W.: Multi-grid Methods and Applications. Springer-Verlag, New York, 1985Google Scholar
  18. 18.
    Hackbusch, W.: Elliptic Differential Equations. Springer-Verlag, New York, 1992Google Scholar
  19. 19.
    Hackbusch, W.: Parabolic multigrid methods. In: R. Glowinski, J.-L. Lions (eds.), Computing Methods in Applied Sciences and Engineering VI, North-Holland, Amsterdam, 1984Google Scholar
  20. 20.
    Hoffmann, K.-H., Tang, Q.: Ginzburg-Landau Phase Transition Theory and Superconductivity. Vol. 134, Birkhäuser, 2001Google Scholar
  21. 21.
    Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic partial differential equations. SIAM J. Sci. Comput. 16(4), 848–864 (1995)MATHGoogle Scholar
  22. 22.
    Kopell, N., Howard, L.N.: Plane wave solutions to reaction-diffusion equations. Stud. Appl. Math. LII(4), 291–328 (1973)Google Scholar
  23. 23.
    Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer-Verlag, 1984Google Scholar
  24. 24.
    Kuramoto, Y., Koga, S.: Turbulized rotating chemical waves. Prog. Theor. Phys. 66(3), 1081–1085 (1981)Google Scholar
  25. 25.
    Larsson, S.: Lecture Notes on Semilinear Parabolic Problems. Department of Mathematics, Chalmers University of Technology, 1996Google Scholar
  26. 26.
    Larsson, S.: A shadowing result with applications to finite element approximation of reaction-diffusion equations. To appear in IMA J. Numer. Anal.Google Scholar
  27. 27.
    Larsson, S., Sanz-Serna, J.-M.: A shadowing result with applications to finite element approximation of reaction-diffusion equations. Math. Comput. 68, 55–72 (1999)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Larsson, S., Thomée, V., Zhou, S.Z.: On multigrid methods for parabolic problems. J. Comput. Math 13, 193–205 (1995)MathSciNetMATHGoogle Scholar
  29. 29.
    Levermore, C.D., Oliver, M.: The complex Ginzburg-Landau equation as a model problem. Lectures in Applied Mathematics, Vol. 31, AMS, Providence, 1996, pp. 141–190Google Scholar
  30. 30.
    Lord, G.L., Stuart, A.M.: Discrete Gevrey regularity, attractors and upper semicontinuity for a finite difference approximation to the Ginzburg-Landau equation. Numer. Func. Anal. Opt. 16, 1003–1047 (1995)MathSciNetMATHGoogle Scholar
  31. 31.
    Lord, G.L.: Attractors and inertial manifolds for the finite-difference approximations of the complex Ginzburg-Landau equation. SIAM J. Numer. Anal. 34(4), 1483–1512 (1997)CrossRefMATHGoogle Scholar
  32. 32.
    Lubich, C., Ostermann, A.: Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization. Numer. Math. 81, 53–84 (1998)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.-L.: Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44(3), 311–370 (2002)MATHGoogle Scholar
  34. 34.
    Morton, K.W., Mayers, D.F.: Numerical Solution of Partial Differential Equations. Cambridge University Press, 1994Google Scholar
  35. 35.
    Murray, J.D.: Mathematical Biology. Springer-Verlag, 1993Google Scholar
  36. 36.
    Ostermann, A., Palencia, C.: Shadowing for nonautonomous parabolic problems with applications to long-time error bounds. SIAM J. Numer. Anal. 37(5), 1399–1419 (2000)CrossRefMATHGoogle Scholar
  37. 37.
    Paullet, J., Ermentrout, B., Troy, W.: The Existence of spiral waves in an oscillatory reaction-diffusion system. SIAM J. Appl. Math. 54(5), 1386–1401 (1994)MATHGoogle Scholar
  38. 38.
    Sherratt, J.A.: On the evolution of periodic plane waves in reaction-diffusion systems of λ - ω type. SIAM J. Appl. Math. 54(5), 1374–1385 (1994)MATHGoogle Scholar
  39. 39.
    Süli, E.: Finite Elements Methods for Partial Differential Equations. Lecture Notes, Oxford University Computing Laboratory, Oxford, 2001Google Scholar
  40. 40.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, 1997Google Scholar
  41. 41.
    Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, London, 2001Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institut für MathematikKarl-Franzens-Universität GrazGrazAustria

Personalised recommendations