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A Samarskii domain decomposition method for two-dimensional convection–diffusion equations

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Abstract

A new Samarskii domain decomposition method for solving two-dimensional convection–diffusion equations is proposed. In this procedure, interface values between subdomains are computed by the Saul’yev asymmetric difference schemes based on Samarskii scheme; interior values are calculated by the Samarskii scheme. The new algorithm obtains accuracy of Samarskii scheme while maintaining parallelism and unconditional stability. Numerical examples show the accuracy and parallel efficiency of the new algorithm.

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References

  • Andreev VB, Savin IA (1995) On the convergence, uniform with respect to the small parameter, of A.A. Samarskii’s monotone scheme and its modifications. Zh. Vychisl. Mat. Mat. Fiz. 35(5), 739–752

  • Chen C, Liu W, Bi C (2013) A two-grid characteristic finite volume element method for semilinear advection-dominated diffusion equations. Numer Methods Partial Differ Equ 29:1543–1562

    Article  MathSciNet  Google Scholar 

  • Dawson CN, Du Q, Dupont TF (1991) A finite difference domain decomposition algorithm for numerical solution of the heat equation. Math Comput 57:63–71

    Article  MathSciNet  Google Scholar 

  • Dryja M, Tu X (2007) A domain decomposition discretization of parabolic problems. Numer Math 107:625–640

    Article  MathSciNet  Google Scholar 

  • Du Q, Mu M, Wu Z (2001) Efficient parallel algorithms for parabolic problems. SIAM J Numer Anal 39(5):1469–1487

    Article  MathSciNet  Google Scholar 

  • Evans DJ (1985) Alternating group explicit method for the diffusion equation. Appl Math Model 9:201–206

    Article  MathSciNet  Google Scholar 

  • Evans DJ, Abdullah ARB (1985) A new explicit method for the diffusion–convection equation. Comput Math Appl 11:145–154

    Article  MathSciNet  Google Scholar 

  • Guo G, Liu B (2013) Unconditional stability of alternating difference schemes with intrinsic parallelism for the fourth-order parabolic equation. Appl Math Comput 219:7319–7328

    Article  MathSciNet  Google Scholar 

  • Guo G, Lü S (2016) Unconditional stability of alternating difference schemes with intrinsic parallelism for two-dimensional fourth-order diffusion equation. Comput Math Appl 71:1944–1959

    Article  MathSciNet  Google Scholar 

  • Guo G, Zhai Y, Liu B (2013) The alternating segment explicit-implicit scheme for the fourth-order parabolic equation. J Inf Comput Sci 10:2981–2991

    Article  Google Scholar 

  • Guo G, Lü S, Liu B (2015) Unconditional stability of alternating difference schemes with variable time steplengthes for dispersive equation. Appl Math Comput 262:249–259

    Article  MathSciNet  Google Scholar 

  • Hao W, Zhu S (2013) Domain decomposition schemes with high-order accuracy and unconditional stability. Appl Math Comput 219:6170–6181

    Article  MathSciNet  Google Scholar 

  • Jia D, Sheng Z, Yuan G (2018) A conservative parallel difference method for 2-dimension diffusion equation. Appl Math Lett 78:72–78

    Article  MathSciNet  Google Scholar 

  • Li C, Yuan Y (2009) A modified upwind difference domain decomposition method for convection–diffusion equations. Appl Numer Math 59:1584–1598

    Article  MathSciNet  Google Scholar 

  • Li C, Yuan Y (2012) Nonoverlapping domain decomposition characteristic finite differences for three-dimensional convection–diffusion equations. Numer Methods Partial Differ Equ 28:17–37

    Article  MathSciNet  Google Scholar 

  • Liang D, Du C (2014) The efficient S-DDM scheme and its analysis for solving parabolic equations. J Comput Phys 272:46–69

    Article  MathSciNet  Google Scholar 

  • Lu J, Zhang B, Xu T (1998) Alternating segment explicit–implicit method for the convection–diffusion equation. Chin J Numer Methods Comput Appl 3:161–167

    MATH  Google Scholar 

  • Ma Y, Sun CP Haake DA, Churchill BM, Ho CM (2012) A high-order alternating direction implicit method for the unsteady convection-dominated diffusion problem. Int J Numer Methods Fluids 70(6):703–712

  • Saul’yev VK (1964) Integration of equations of parabolic type by method of nets. Macmillan Company, New York

    MATH  Google Scholar 

  • Sheng Z, Yuan G, Hang X (2007) Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation. Appl Math Comput 184(2):1015–1031

    Article  MathSciNet  Google Scholar 

  • Shi H, Liao H (2006) Unconditional stability of corrected explicit–implicit domain decomposition algorithms for parallel approximation of heat equations. SIAM J Numer Anal 44:1584–1611

    Article  MathSciNet  Google Scholar 

  • Wang W (2004) A class of alternating segment Crank–Nicolson methods for solving convection–diffusion equations. Computing 73:41–55

    Article  MathSciNet  Google Scholar 

  • Wang W, Fu S (2006) An unconditionally stable alternating segment difference scheme of eight points for the dispersive equation. Int J Numer Methods Eng 67:435–447

    Article  MathSciNet  Google Scholar 

  • Xue G, Feng H (2018) New parallel algorithm for convection-dominated diffusion equation. East Asian J Appl Math 8(2):261–279

    Article  MathSciNet  Google Scholar 

  • Xue G, Feng H (2020) An alternating segment explicit-implicit scheme with intrinsic parallelism for Burgers’ equation. J Comput Theor Trans 49(1):15–30

    Article  MathSciNet  Google Scholar 

  • Yuan G, Shen L, Zhou Y (1999) Unconditional stability of alternating difference schemes with intrinsic parallelism for two-dimensional parabolic systems. Numer Methods Partial Differ Equ 15:625–636

    Article  MathSciNet  Google Scholar 

  • Yuan G, Sheng Z, Hang X (2007) The unconditional stability of parallel difference schemes with scheme second order convergence for nonlinear parabolic system. J Partial Differ Equ 20(1):45–64

    MathSciNet  MATH  Google Scholar 

  • Zhang Q, Wang W (2009) A four-order alternating segment Crank–Nicolson scheme for the dispersive equation. Comput Math Appl 57:283–289

    Article  MathSciNet  Google Scholar 

  • Zhou Z, Liang D (2017) The mass-preserving and modified-upwind splitting DDM scheme for time-dependent convection–diffusion equations. J Comput Appl Math 317:247–273

    Article  MathSciNet  Google Scholar 

  • Zhou Z, Liang D (2018) Mass-preserving time second-order explicit–implicit domain decomposition schemes for solving parabolic equations with variable coefficients. Comput Appl Math 37:4423–4442

    Article  MathSciNet  Google Scholar 

  • Zhou Z, Liang D, Wong Y (2018) The new mass-conserving S-DDM scheme for two-dimensional parabolic equations with variable coefficients. Appl Math Comput 338:882–902

    Article  MathSciNet  Google Scholar 

  • Zhou Z, Sun X, Pan H, Wang Y (2020) An efficient characteristic finite difference S-DDM scheme for convection–diffusion equations. Comput Math Appl 80:3044–3065

    Article  MathSciNet  Google Scholar 

  • Zhu S (2010) Conservative domain decomposition procedure with unconditional stability and second-order accuracy. Appl Math Comput 216:3275–3282

    Article  MathSciNet  Google Scholar 

  • Zhu L, Yuan G, Du Q (2010) An efficient explicit/implicit domain decomposition method for convection–diffusion equations. Numer Methods Partial Differ Equ 26:852–873

  • Zhuang Y, Sun X (2002) Stabilized explicit–implicit domain decomposition methods for the numerical solution of parabolic equations. SIAM J Sci Comput 24:335–358

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank referees for their comments and suggestions which have helped to improve the paper. This work was supported by National Natural Science Foundation of China (no. 12101536), PhD research startup foundation of Yantai University (no. 2219002) and LCP Fund for Young Scholar. The authors thank Prof. Guangwei Yuan and Dr. Pengcheng Mu for interesting discussions on the subject.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Guanyu Xue

  2. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Guanyu Xue

  3. School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China

    Yulong Gao

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  1. Guanyu Xue
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  2. Yulong Gao
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Correspondence to Guanyu Xue.

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Communicated by Abimael Loula.

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Xue, G., Gao, Y. A Samarskii domain decomposition method for two-dimensional convection–diffusion equations. Comp. Appl. Math. 41, 283 (2022). https://doi.org/10.1007/s40314-022-01986-0

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  • DOI: https://doi.org/10.1007/s40314-022-01986-0

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