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W-Methods and Approximate Matrix Factorization for Parabolic PDEs with Mixed Derivative Terms

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Rosenbrock—Wanner–Type Methods

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Abstract

In this chapter W-methods are combined with the Approximate Matrix Factorization technique (AMF) in alternating direction implicit (ADI) sense for the time integration of parabolic partial differential equations with mixed derivatives in the elliptic operator, previously discretized in space by means of Finite Differences. Three different families of AMF-type W-methods are introduced and their unconditional stability is analized regardless of the spatial dimension. To this aim, a scalar test equation is presented and it is shown to be relevant for the class of problems under consideration when either periodic or homogeneous Dirichlet boundary conditions are imposed. Numerical results comparing the proposed AMF-type W-methods and some classical ADI schemes in the literature for 2 ≤ m ≤ 4 space dimensions are presented.

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References

  1. I.J.D. Craig, A.D. Sneyd, An alternating-direction implicit scheme for parabolic equations with mixed derivatives. Comput. Math. Appl. 16(4), 341–350 (1988)

    Article  MathSciNet  Google Scholar 

  2. J. Douglas, Jr., H.H. Rachford, Jr., On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)

    Article  MathSciNet  Google Scholar 

  3. B. Düring, M. Fournié, A. Rigal, High-order ADI schemes for convection-diffusion equations with mixed derivative terms, in Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM’12, ed. by T. Gammarth, M. Azaïez, et al. Lecture Notes in Computational Science and Engineering, vol. 95 (Springer, Berlin, 2014), pp. 217–226

    Google Scholar 

  4. B. Düring, J. Miles, High-order ADI scheme for option pricing in stochastic volatility models. J. Comput. Appl. Math. 316, 109–121 (2017)

    Article  MathSciNet  Google Scholar 

  5. A. Gerisch, J.G. Verwer, Operator splitting and approximate factorization for taxis-diffusion-reaction models. Appl. Numer. Math. 42(1–3), 159–176 (2002). Ninth Seminar on Numerical Solution of Differential and Differential-Algebraic Equations (Halle, 2000)

    Google Scholar 

  6. S. González Pinto, D. Hernández Abreu, S. Pérez Rodríguez, Rosenbrock-type methods with inexact AMF for the time integration of advection diffusion reaction PDEs. J. Comput. Appl. Math. 262, 304–321 (2014)

    Article  MathSciNet  Google Scholar 

  7. S. González Pinto, D. Hernández Abreu, S. Pérez Rodríguez, R. Weiner, A family of three-stage third order AMF-W-methods for the time integration of advection diffusion reaction PDEs. Appl. Math. Comput. 274, 565–584 (2016)

    MathSciNet  MATH  Google Scholar 

  8. S. González Pinto, D. Hernández Abreu, S. Pérez Rodríguez, W-methods to stabilize standard explicit Runge-Kutta methods in the time integration of advection-diffusion-reaction PDEs. J. Comput. Appl. Math. 316, 143–160 (2017)

    Article  MathSciNet  Google Scholar 

  9. S. González Pinto, E. Hairer, D. Hernández Abreu, S. Pérez Rodríguez, AMF-type W-methods for parabolic PDEs with mixed derivatives. SIAM J. Sci. Comput. 40(5), A2905–A2929 (2018)

    Article  Google Scholar 

  10. S. González Pinto, E. Hairer, D. Hernández Abreu, S. Pérez Rodríguez, PDE-W-methods for parabolic problems with mixed derivatives. Numer. Algorithms 78, 957–981 (2018)

    Article  MathSciNet  Google Scholar 

  11. E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14, 2nd edn. (Springer, Berlin, 1996)

    Google Scholar 

  12. E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations I. Non Stiff Problems. Springer Series in Computational Mathematics, vol. 8, 2nd rev. edn. (Springer, Berlin, 1993)

    Google Scholar 

  13. C. Hendricks, M. Ehrhardt, M. Günther, High-order ADI schemes for diffusion equations with mixed derivatives in the combination technique. Appl. Numer. Math. 101, 36–52 (2016)

    Article  MathSciNet  Google Scholar 

  14. C. Hendricks, C. Heuer, M. Ehrhardt, M. Günther, High-order ADI finite difference schemes for parabolic equations in the combination technique with application in finance. J. Comput. Appl. Math. 316, 175–194 (2017)

    Article  MathSciNet  Google Scholar 

  15. S.L. Heston, A closed form solution for options with stochastic volatility with applications to bonds and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)

    Article  MathSciNet  Google Scholar 

  16. W. Hundsdorfer, Accuracy and stability of splitting with stabilizing corrections. Appl. Numer. Math. 42(1–3), 213–233 (2002). Ninth Seminar on Numerical Solution of Differential and Differential-Algebraic Equations (Halle, 2000)

    Google Scholar 

  17. W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33 (Springer, Berlin, 2003)

    Google Scholar 

  18. K.J. in ’t Hout, S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Model. 7(2), 303–320 (2010)

    Google Scholar 

  19. K.J. in ’t Hout, B.D. Welfert, Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms. Appl. Numer. Math. 59(3–4), 677–692 (2009)

    Google Scholar 

  20. K.J. in ’t Hout, M. Wyns, Convergence of the modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with mixed derivative term. J. Comput. Appl. Math. 296, 170–180 (2016)

    Google Scholar 

  21. T. Jax, G. Steinebach, Generalized ROW-type methods for solving semi-explicit DAEs of index-1. J. Comput. Appl. Math. 316, 213–228 (2017)

    Article  MathSciNet  Google Scholar 

  22. J. Lang, J.G. Verwer, W-methods in optimal control. Numer. Math. 124(2), 337–360 (2013)

    Article  MathSciNet  Google Scholar 

  23. D.W. Peaceman, H.H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl. Math. 3, 28–41 (1955)

    Article  MathSciNet  Google Scholar 

  24. A. Rahunanthan, D. Stanescu, High-order W-methods. J. Comput. Appl. Math. 233, 1798–1811 (2010)

    Article  MathSciNet  Google Scholar 

  25. J. Rang, L. Angermann, New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1. BIT Numer. Math. 45(4), 761–787 (2005)

    Article  MathSciNet  Google Scholar 

  26. W. Schoutens, E. Simons, J. Tistaert, A perfect calibration! Now what? Wilmott Mag. 66–78 (2004)

    Google Scholar 

  27. T. Steihaug, A. Wolfbrandt, An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations. Math. Comput. 33(146), 521–534 (1979)

    Article  MathSciNet  Google Scholar 

  28. P.J. van der Houwen, B.P. Sommeijer, Approximate factorization for time-dependent partial differential equations. J. Comput. Appl. Math. 128(1–2), 447–466 (2001). Numerical analysis 2000, Vol. VII, Partial differential equations

    Google Scholar 

  29. G. Winkler, T. Apel, U. Wystup, Valuation of options in Heston’s stochastic volatility model using finite element methods, in Foreign Exchange Risk, ed. by J. Hakala, U. Wystup (Risk Publ., London, 2002)

    Google Scholar 

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Acknowledgements

The authors thank Ernst Hairer and Soledad Pérez-Rodríguez for the revision and their scientific contribution to the content of the current chapter, which is collected in references [9, 10].

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, La Laguna, Spain

    Severiano González-Pinto & Domingo Hernández-Abreu

Authors
  1. Severiano González-Pinto
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  2. Domingo Hernández-Abreu
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Corresponding author

Correspondence to Severiano González-Pinto .

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Editors and Affiliations

  1. Fachbereich Elektrotechnik. Maschinenbau und Technikjournalismus, Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Nordrhein-Westfalen, Germany

    Tim Jax

  2. Arbeitsgruppe Angewandte Mathematik und Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Andreas Bartel

  3. Arbeitsgruppe Angewandte Mathematik und Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Matthias Ehrhardt

  4. Arbeitsgruppe Angewandte Mathematik und Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Michael Günther

  5. Fachbereich Elektrotechnik. Maschinenbau und Technikjournalismus und Institut für Technik, Ressourcenschonung und Energieeffizienz, Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Nordrhein-Westfalen, Germany

    Gerd Steinebach

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González-Pinto, S., Hernández-Abreu, D. (2021). W-Methods and Approximate Matrix Factorization for Parabolic PDEs with Mixed Derivative Terms. In: Jax, T., Bartel, A., Ehrhardt, M., Günther, M., Steinebach, G. (eds) Rosenbrock—Wanner–Type Methods. Mathematics Online First Collections. Springer, Cham. https://doi.org/10.1007/978-3-030-76810-2_4

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