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A splitting preconditioner for implicit Runge-Kutta discretizations of a partial differential-algebraic equation

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Abstract

In this paper we study efficient iterative methods for solving the system of linear equations arising from the fully implicit Runge-Kutta discretizations of a class of partial differential-algebraic equations. In each step of the time integration, a block two-by-two linear system is obtained and needed to be solved numerically. A preconditioning strategy based on an alternating Kronecker product splitting of the coefficient matrix is proposed to solve such linear systems. Some spectral properties of the preconditioned matrix are established and numerical examples are presented to demonstrate the effectiveness of this approach.

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References

  1. Bai, Z.Z., Golub, G.H., Pan, J.Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benzi, M., Golub, G.H.: A preconditioner for generalized saddle point problems. SIAM J. Matrix. Anal. Appl. 26, 20–41 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berman, A., Plemmons, R.J.: Nonnegative matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  4. Briggs, W., Henson, V.E., McCormick, S.F.: A multigrid tutorial. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  5. Chen, H.: A splitting preconditioner for the iterative solution of implicit Runge-Kutta and boundary value methods. BIT 54, 607–621 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, H.: Generalized Kronecker product splitting iteration for the solution of implicit Runge-Kutta and boundary value methods. Numer. Linear. Algebra. Appl. 22, 357–370 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dax, A.: The convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations. SIAM Rev. 32, 611–635 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Colli Franzone, P., Deuflhard, P., Erdmann, B., Lang, J., Pavarino, L.F.: Adaptivity in space and time for reaction-diffusion systems in electrocardiology. SIAM J. Sci. Comput. 28, 942–962 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Elman, H.C., Ramage, A., Silvester, D.J.: IFISS: a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Software 33, Article 14 (2007)

    Article  MathSciNet  Google Scholar 

  10. Elman, H.C., Silvester, D.J., Wathen, A.: Finite elements and fast iterative solvers with applications in incompressible fluid dynamics. Numer. Math. Sci. Comput. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  11. Ethier, M., Bourgault, Y.: Semi-implicit time-discretization schemes for the Bidomain model. SIAM J. Numer. Anal. 46, 2443–2468 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gerardo-Giorda, L., Mirabella, L., Nobile, F., Perego, M., Veneziani, A.: A model-based block-triangular preconditioner for the Bidomain system in electrocardiology. J. Comput. Phys. 228, 3625–3639 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gerardo-Giorda, L., Mirabella, L.: Spectral analysis of a block-triangular preconditioner for the Bidomain system in electrocardiology. Electron. Trans. Numer. Anal. 39, 186–201 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Hairer, E., Wanner, G.: Solving ordinary differential equations II. Stiff and differential algebraic problems. Springer, Berlin (1996)

    MATH  Google Scholar 

  15. Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  16. Keener, J., Sneyd, J.: Mathematical physiology, 2nd ed., vol. I-II. Springer, New York (2009)

    Book  MATH  Google Scholar 

  17. Mardal, K.A., Nielsen, B.F., Cai, X., Tveito, A.: An order optimal solver for the discretized bidomain equations. Numer. Linear. Algebra. Appl. 14, 83–98 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marsh, M.E., Ziaratqahi, S.T., Spiteri, R.J.: The secrets to the success of the Rush-Larsen method and its generalizations. IEEE Trans. Biomed. Eng. 59, 2506–2515 (2012)

    Article  Google Scholar 

  19. Nilssen, T.K., Staff, G.A., Mardal, K.A.: Order-optimal preconditioners for fully implicit Runge-Kutta schemes applied to the bidomain equations. Numer. Meth. Part. Diff. Equ. 27, 1290–1312 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pavarino, L.F., Scacchi, S.: Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31, 420–443 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pavarino, L.F., Scacchi, S.: Parallel multilevel Schwarz and block preconditioners for the Bidomain parabolic-parobolic and parabolic-elliptic formulations. SIAM J. Sci. Comput. 33, 1897–1919 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pennacchio, M., Simoncini, V.: Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145, 49–70 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pennacchio, M., Simoncini, V.: Algebraic multigrid preconditioners for the bidomain reaction-diffusion system. Appl. Numer. Math. 59, 3033–3050 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pennacchio, M., Simoncini, V.: Fast structured AMG preconditioning for the bidomain model in electrocadiology. SIAM J. Sci. Comput. 33, 721–745 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Perego, M., Veneziani, A.: An efficient generalization of the Rush-Larsen method for solving electro-physiology membrane equations. Electron. Trans. Numer. Anal. 35, 234–256 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Plank, G., Liebmann, M., Weber Dos Santos, R., Vigmond, E.J., Haase, G.: Algebraic multigrid preconditioner for the cardiac Bidomain model. IEEE Trans. Biomed. Engrg. 54, 585–596 (2007)

    Article  Google Scholar 

  27. Ruge, J.W., Stüben, K.: Algebraic multigrid. In: McCormick, S.F. (ed.) Multigrid methods, Frontiers Appl. Math., vol. 3, pp 73–130. SIAM, Philadelphia (1987)

  28. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Simoncini, V., Benzi, M.: Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems. SIAM J. Matrix Anal. Appl. 26, 377–389 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sundnes, J., Lines, G.T., Mardal, K.A., Tveito, A.: Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart. Comput. Methods Biomech. Biomed. Engrg. 5, 397–409 (2002)

    Article  Google Scholar 

  32. Sundnes, J., Lines, G.T., Tveito, A.: An operator splitting method for solving the Bidomain equations coupled to a volume conductor model for the torso. Comput. Math. Biosci. 194, 233–248 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Varga, R.S.: Matrix iterative analysis. Prentice-Hall, Englewood Cliffs (1962)

    Google Scholar 

  34. Vigmond, E.J., Weber dos Santos, R., Prassl, A.J., Deo, M., Plank, G.: Solvers for the cardiac bidomain equations. Progr. Biophys. Molecular Biol. 96, 3–18 (2008)

    Article  Google Scholar 

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  1. College of Mathematics Science, Chongqing Normal University, Chongqing, 401331, China

    Hao Chen

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  1. Hao Chen
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Correspondence to Hao Chen.

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Chen, H. A splitting preconditioner for implicit Runge-Kutta discretizations of a partial differential-algebraic equation. Numer Algor 73, 1037–1054 (2016). https://doi.org/10.1007/s11075-016-0128-5

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  • DOI: https://doi.org/10.1007/s11075-016-0128-5

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