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A Dimension Splitting Method for Time Dependent PDEs on Irregular Domains

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Abstract

We develop a simple and efficient dimension splitting method for solving time dependent partial differential equations (PDEs) on multiple space dimensional irregular domains using a one dimensional kernel-free boundary integral (KFBI) method. The proposed method extends the alternating direction implicit methods and locally one dimensional methods to more general cases involving complex geometry. The KFBI method is a potential theory based Cartesian grid method, which works as an improvement of conventional boundary integral methods. In the KFBI method, boundary or volume integrals are evaluated by solving equivalent interface problems without using any analytic expression of Green’s functions. The one dimensional interface problems after dimension splitting are solved by finite difference method. The resulting linear systems are tri-diagonal and efficiently solved by the Thomas algorithm. The one dimensional kernel-free boundary integral method is rigorously proved to have second-order convergence rate in the maximum norm. Multiple numerical examples, including different types of PDEs and a free boundary problem, are presented to demonstrate the advantages of the proposed method. Numerical results show that the proposed method is efficient and achieves overall second order accuracy.

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Data availibility

The custom code of the current study is available at https://github.com/zhouhan-sjtu/KFBI-OS. The data generated in this study are available upon reasonable request.

References

  1. Beale, J.T.: A grid-based boundary integral method for elliptic problems in three dimensions. SIAM J. Numer. Anal. 42(2), 599–620 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, S., Merriman, B., Osher, S., Smereka, P.: A simple level set method for solving Stefan problems. J. Comput. Phys. 135(1), 8–29 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM (2002)

  4. Douglas, J., Gunn, J.E.: Alternating direction methods for parabolic systems in mspace variables. J. ACM (JACM) 9(4), 450–456 (1962)

    Article  MATH  Google Scholar 

  5. Douglas, J., Jr.: On the numerical integration of \(\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}\) by implicit methods. J. Soc. Ind. Appl. Math. 3(1), 42–65 (1955)

    Article  Google Scholar 

  6. D’Yakonov, E.: Difference schemes with splitting operators for multidimensional unsteady problems (English translation). URSS Comp. Math. 3, 581–607 (1963)

    Google Scholar 

  7. Geiser, J.: Operator splitting methods for wave equations. Int. Math. Forum 2 (2007)

  8. Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Courier Corporation (2012)

  9. Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, vol. 33. Springer (2007)

  10. Kim, S., Lim, H.: High-order schemes for acoustic waveform simulation. Appl. Numer. Math. 57(4), 402–414 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kress, R.: Linear Integral Equations, vol. 82. Springer (1989)

  12. LeVeque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31(4), 1019–1044 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, Z., Mayo, A.: ADI Methods for Heat Equations with Discontinuities Along an Arbitrary Interface. IBM Thomas J, Watson Research Division (1993)

  14. Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35(1), 230–254 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, J., Zheng, Z.: A dimension by dimension splitting immersed interface method for heat conduction equation with interfaces. J. Comput. Appl. Math. 261, 221–231 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, J.K., Zheng, Z.S.: Efficient high-order immersed interface methods for heat equations with interfaces. Appl. Math. Mech. (Engl. Ed.) 35(9), 1189–1202 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ma, C., Zhang, Q., Zheng, W.: A high-order fictitious-domain method for the advection-diffusion equation on time-varying domain. arXiv e-prints arXiv:2104.01870 (2021)

  18. Mayo, A.: The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21(2), 285–299 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Peaceman, D.W., Rachford, H.H., Jr.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  21. Peskin, C.S.: The immersed boundary method. Acta Numerica 11, 479–517 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM (2003)

  23. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wei, Z., Li, C., Zhao, S.: A spatially second order alternating direction implicit (adi) method for solving three dimensional parabolic interface problems. Comput. Math. Appl. 75(6), 2173–2192 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xie, Y., Ying, W.: A fourth-order kernel-free boundary integral method for the modified Helmholtz equation. J. Sci. Comput. 78(3), 1632–1658 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xie, Y., Ying, W.: A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions. J. Comput. Phys. 415, 109526 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xie, Y., Ying, W.: A high-order kernel-free boundary integral method for incompressible flow equations in two space dimensions. Numer. Math. 13(3), 595–619 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Xie, Y., Ying, W., Wang, W.C.: A high-order kernel-free boundary integral method for the biharmonic equation on irregular domains. J. Sci. Comput. 80(3), 1681–1699 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yanenko, N.N.: On the convergence of the splitting method for the heat conductivity equation with variable coefficients. USSR Comput. Math. Math. Phys. 2(5), 1094–1100 (1963)

    Article  MATH  Google Scholar 

  30. Ying, W.: A cartesian grid-based boundary integral method for an elliptic interface problem on closely packed cells. Commun. Comput. Phys. 24(4), 1196–1220 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ying, W., Henriquez, C.S.: A kernel-free boundary integral method for elliptic boundary value problems. J. Comput. Phys. 227(2), 1046–1074 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ying, W., Wang, W.C.: A kernel-free boundary integral method for implicitly defined surfaces. J. Comput. Phys. 252, 606–624 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ying, W., Wang, W.C.: A kernel-free boundary integral method for variable coefficients elliptic pdes. Commun. Comput. Phys. 15(4), 1108–1140 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhao, S.: A matched alternating direction implicit (adi) method for solving the heat equation with interfaces. J. Sci. Comput. 63(1), 118–137 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhou, H., Ying, W.: https://github.com/zhouhan-sjtu/KFBI-OS (2022)

  36. Zhou, Y.C., Zhao, S., Feig, M., Wei, G.W.: High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213(1), 1–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals. Elsevier (2005)

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Funding

This work is financially supported by the National Key R & D Program of China, Project Number 2020YFA0712000. It is also partially supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010405), the National Natural Science Foundation of China (Grant No. DMS-11771290) and the Science Challenge Project of China (Grant No. TZ2016002).

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Correspondence to Wenjun Ying.

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This work is financially supported by the National Key R &D Program of China, Project Number 2020YFA0712000. It is also partially supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010405), the National Natural Science Foundation of China (Grant No. DMS-11771290) and the Science Challenge Project of China (Grant No. TZ2016002).

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Zhou, H., Ying, W. A Dimension Splitting Method for Time Dependent PDEs on Irregular Domains. J Sci Comput 94, 20 (2023). https://doi.org/10.1007/s10915-022-02066-5

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