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Erschienen in: Numerical Algorithms 2/2020

10.07.2019 | Original Paper

Preconditioned quasi-compact boundary value methods for space-fractional diffusion equations

verfasst von: Yongtao Zhou, Chengjian Zhang, Luigi Brugnano

Erschienen in: Numerical Algorithms | Ausgabe 2/2020

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Abstract

This paper focuses on highly efficient numerical methods for solving space-fractional diffusion equations. By combining the fourth-order quasi-compact difference scheme and boundary value methods, a class of quasi-compact boundary value methods are constructed. In order to accelerate the convergence rate of this class of methods, the Kronecker product splitting (KPS) iteration method and the preconditioned method with KPS preconditioner are proposed. A convergence criterion for the KPS iteration method is derived. A numerical experiment further illustrates the computational efficiency and accuracy of the proposed methods. Moreover, a numerical comparison with the preconditioned method with Strang-type preconditioner is given, which shows that the preconditioned method with KPS preconditioner is comparable in computational efficiency.
Literatur
1.
Zurück zum Zitat Amodio, P., Brugnano, L.: The conditioning of Toeplitz banded matrices. Math. Comput. Modelling 23, 29–42 (1996) MathSciNetMATH Amodio, P., Brugnano, L.: The conditioning of Toeplitz banded matrices. Math. Comput. Modelling 23, 29–42 (1996) MathSciNetMATH
2.
Zurück zum Zitat Bai, J., Feng, X.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16, 2492–2502 (2007) MathSciNet Bai, J., Feng, X.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16, 2492–2502 (2007) MathSciNet
3.
Zurück zum Zitat Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000) Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)
4.
Zurück zum Zitat Brugnano, L.: Essentially symplectic boundary value methods for linear Hamiltonian systems. J. Comput. Math. 15, 233–252 (1997) MathSciNetMATH Brugnano, L.: Essentially symplectic boundary value methods for linear Hamiltonian systems. J. Comput. Math. 15, 233–252 (1997) MathSciNetMATH
5.
Zurück zum Zitat Brugnano, L., Trigiante, D.: Tridiagonal matrices: invertibility and conditioning. Linear Algebra Appl. 166, 131–150 (1992) MathSciNetMATH Brugnano, L., Trigiante, D.: Tridiagonal matrices: invertibility and conditioning. Linear Algebra Appl. 166, 131–150 (1992) MathSciNetMATH
6.
Zurück zum Zitat Brugnano, L., Trigiante, D.: Convergence and stability of boundary value methods for ordinary differential equations. J. Comput. Appl. Math. 66, 97–109 (1996) MathSciNetMATH Brugnano, L., Trigiante, D.: Convergence and stability of boundary value methods for ordinary differential equations. J. Comput. Appl. Math. 66, 97–109 (1996) MathSciNetMATH
7.
Zurück zum Zitat Brugnano, L., Trigiante, D.: Solving differential equations by multistep initial and boundary value methods. Gordan and Breach, Amsterdam (1998) MATH Brugnano, L., Trigiante, D.: Solving differential equations by multistep initial and boundary value methods. Gordan and Breach, Amsterdam (1998) MATH
8.
Zurück zum Zitat Brugnano, L., Trigiante, D.: Boundary value methods: the third way between linear multistep and Runge-Kutta methods. Comput. Math. Appl. 36, 269–284 (1998) MathSciNetMATH Brugnano, L., Trigiante, D.: Boundary value methods: the third way between linear multistep and Runge-Kutta methods. Comput. Math. Appl. 36, 269–284 (1998) MathSciNetMATH
9.
Zurück zum Zitat Brugnano, L., Zhang, C., Li, D.: A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schrödinger equation with wave operator. Comm. Nonlinear Sci. Numer. Simmu. 60, 33–49 (2018) Brugnano, L., Zhang, C., Li, D.: A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schrödinger equation with wave operator. Comm. Nonlinear Sci. Numer. Simmu. 60, 33–49 (2018)
10.
Zurück zum Zitat Chan, R.H., Ng, M.K., Jin, X.: Strang-type preconditioners for systems of LMF-based ODE codes. IMA J. Numer. Anal. 21, 451–462 (2001) MathSciNetMATH Chan, R.H., Ng, M.K., Jin, X.: Strang-type preconditioners for systems of LMF-based ODE codes. IMA J. Numer. Anal. 21, 451–462 (2001) MathSciNetMATH
11.
Zurück zum Zitat Chen, H., Zhang, C.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218, 2619–2630 (2011) MathSciNetMATH Chen, H., Zhang, C.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218, 2619–2630 (2011) MathSciNetMATH
12.
Zurück zum Zitat Chen, H., Zhang, C.: Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations. Appl. Numer. Math. 62, 141–154 (2012) MathSciNetMATH Chen, H., Zhang, C.: Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations. Appl. Numer. Math. 62, 141–154 (2012) MathSciNetMATH
13.
Zurück zum Zitat Chen, H.: A splitting preconditioner for the iterative solution of implicit Runge-Kutta and boundary value methods. BIT 54, 607–621 (2014) MathSciNetMATH Chen, H.: A splitting preconditioner for the iterative solution of implicit Runge-Kutta and boundary value methods. BIT 54, 607–621 (2014) MathSciNetMATH
14.
Zurück zum Zitat Chen, H.: Kronecker product splitting preconditioners for implicit Runge-Kutta discretizations of viscous wave equations. Appl. Math. Model. 40, 4429–4440 (2016) MathSciNetMATH Chen, H.: Kronecker product splitting preconditioners for implicit Runge-Kutta discretizations of viscous wave equations. Appl. Math. Model. 40, 4429–4440 (2016) MathSciNetMATH
15.
Zurück zum Zitat Chen, H.: A splitting preconditioner for implicit Runge-Kutta discretizations of a partial differential-algebraic equation. Numer. Algor. 73, 1037–1054 (2016) MathSciNetMATH Chen, H.: A splitting preconditioner for implicit Runge-Kutta discretizations of a partial differential-algebraic equation. Numer. Algor. 73, 1037–1054 (2016) MathSciNetMATH
16.
Zurück zum Zitat Chen, H., Zhang, T., Lv, W.: Block preconditioning strategies for time-space fractional diffusion equations. Appl. Math. Comput. 337, 41–53 (2018) MathSciNetMATH Chen, H., Zhang, T., Lv, W.: Block preconditioning strategies for time-space fractional diffusion equations. Appl. Math. Comput. 337, 41–53 (2018) MathSciNetMATH
17.
Zurück zum Zitat Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014) MathSciNetMATH Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014) MathSciNetMATH
18.
Zurück zum Zitat Chen, M., Deng, W.: Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators. Comm. Comput. Phys. 16, 516–540 (2014) MathSciNetMATH Chen, M., Deng, W.: Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators. Comm. Comput. Phys. 16, 516–540 (2014) MathSciNetMATH
19.
Zurück zum Zitat Davis, P.: Circulant matrices. AMS Chelsea Publishing, Rhode Island (1994) MATH Davis, P.: Circulant matrices. AMS Chelsea Publishing, Rhode Island (1994) MATH
20.
Zurück zum Zitat Gal, N., Weihs, D.: Experimental evidence of strong anomalous diffusion in living cells. Phys. Rev. E 81, 020903 (2010) Gal, N., Weihs, D.: Experimental evidence of strong anomalous diffusion in living cells. Phys. Rev. E 81, 020903 (2010)
21.
Zurück zum Zitat Ganti, V., Meerschaert, M.M., Georgiou, E.F., Viparelli, E., Parker, G.: Normal and anomalous diffusion of gravel tracer particles in rivers. J. Geophys. Res. 115, F00A07 (2010) Ganti, V., Meerschaert, M.M., Georgiou, E.F., Viparelli, E., Parker, G.: Normal and anomalous diffusion of gravel tracer particles in rivers. J. Geophys. Res. 115, F00A07 (2010)
22.
Zurück zum Zitat Gu, X., Huang, T., Zhao, X., Li, H., Li, L.: Strang-type preconditioners for solving fractional diffusion equations by boundary value methods. J. Comput. Appl. Math. 277, 73–86 (2015) MathSciNetMATH Gu, X., Huang, T., Zhao, X., Li, H., Li, L.: Strang-type preconditioners for solving fractional diffusion equations by boundary value methods. J. Comput. Appl. Math. 277, 73–86 (2015) MathSciNetMATH
23.
Zurück zum Zitat Hao, Z., Sun, Z., Cao, W.: A fourth-order approximation of fractional derivatives with its applications. J. Comput. Phys. 281, 787–805 (2015) MathSciNetMATH Hao, Z., Sun, Z., Cao, W.: A fourth-order approximation of fractional derivatives with its applications. J. Comput. Phys. 281, 787–805 (2015) MathSciNetMATH
24.
Zurück zum Zitat Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1991) MATH Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1991) MATH
25.
Zurück zum Zitat Iavernaro, F., Mazzia, F.: Convergence and stability of multistep methods solving nonlinear initial value problems. SIAM J. Sci. Comput. 18, 270–285 (1997) MathSciNetMATH Iavernaro, F., Mazzia, F.: Convergence and stability of multistep methods solving nonlinear initial value problems. SIAM J. Sci. Comput. 18, 270–285 (1997) MathSciNetMATH
26.
Zurück zum Zitat Iavernaro, F., Mazzia, F.: Block-boundary value methods for the solution of ordinary differential equations. SIAM J. Sci. Comput. 21, 323–339 (1999) MathSciNetMATH Iavernaro, F., Mazzia, F.: Block-boundary value methods for the solution of ordinary differential equations. SIAM J. Sci. Comput. 21, 323–339 (1999) MathSciNetMATH
27.
Zurück zum Zitat Lei, S., Huang, Y.: Fast algorithms for high-order numerical methods for space fractional diffusion equations. Int. J. Comput. Math. 94, 1062–1078 (2017) MathSciNetMATH Lei, S., Huang, Y.: Fast algorithms for high-order numerical methods for space fractional diffusion equations. Int. J. Comput. Math. 94, 1062–1078 (2017) MathSciNetMATH
28.
Zurück zum Zitat Lei, S., Sun, H.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013) MathSciNetMATH Lei, S., Sun, H.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013) MathSciNetMATH
29.
Zurück zum Zitat Li, C., Zhang, C.: Block boundary value methods applied to functional differential equations with piecewise continuous arguments. Appl. Numer. Math. 115, 214–224 (2017) MathSciNetMATH Li, C., Zhang, C.: Block boundary value methods applied to functional differential equations with piecewise continuous arguments. Appl. Numer. Math. 115, 214–224 (2017) MathSciNetMATH
30.
Zurück zum Zitat Li, C., Zhang, C.: The extended generalized Störmer-Cowell methods for second-order delay boundary value problems. Appl. Math. Comput. 294, 87–95 (2017) MathSciNetMATH Li, C., Zhang, C.: The extended generalized Störmer-Cowell methods for second-order delay boundary value problems. Appl. Math. Comput. 294, 87–95 (2017) MathSciNetMATH
31.
32.
Zurück zum Zitat Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004) MathSciNetMATH Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004) MathSciNetMATH
33.
Zurück zum Zitat Oldham, K.B.: Fractional differential equations in electrochemistry. Adv. Engrg. Software 41, 9–12 (2010) MATH Oldham, K.B.: Fractional differential equations in electrochemistry. Adv. Engrg. Software 41, 9–12 (2010) MATH
34.
Zurück zum Zitat Pang, H., Sun, H.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231, 693–703 (2012) MathSciNetMATH Pang, H., Sun, H.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231, 693–703 (2012) MathSciNetMATH
35.
Zurück zum Zitat Podlubny, I.: Fractional differential equations. Academic Press, New York (1999) MATH Podlubny, I.: Fractional differential equations. Academic Press, New York (1999) MATH
36.
Zurück zum Zitat Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314, 749–755 (2002) MATH Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314, 749–755 (2002) MATH
37.
Zurück zum Zitat Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986) MathSciNetMATH Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986) MathSciNetMATH
38.
Zurück zum Zitat Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: theory and applications. Gordon and Breach, New York (1993) MATH Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: theory and applications. Gordon and Breach, New York (1993) MATH
39.
Zurück zum Zitat Scher, H., Montroll, E.W.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455–2477 (1975) Scher, H., Montroll, E.W.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455–2477 (1975)
40.
Zurück zum Zitat Wang, H., Du, N.: A super fast-preconditioned iterative method for steady-state space-fractional diffusion equations. J. Comput. Phys. 240, 49–57 (2013) MathSciNetMATH Wang, H., Du, N.: A super fast-preconditioned iterative method for steady-state space-fractional diffusion equations. J. Comput. Phys. 240, 49–57 (2013) MathSciNetMATH
41.
Zurück zum Zitat Wang, H., Wang, K., Sircar, T.: A direct \(\mathcal {O}(N\log _{2} {N)}\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010) MathSciNetMATH Wang, H., Wang, K., Sircar, T.: A direct \(\mathcal {O}(N\log _{2} {N)}\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010) MathSciNetMATH
42.
Zurück zum Zitat Wang, H., Zhang, C., Zhou, Y.: A class of compact boundary value methods applied to semi-linear reaction-diffusion equations. Appl. Math. Comput. 325, 69–81 (2018) MathSciNetMATH Wang, H., Zhang, C., Zhou, Y.: A class of compact boundary value methods applied to semi-linear reaction-diffusion equations. Appl. Math. Comput. 325, 69–81 (2018) MathSciNetMATH
43.
Zurück zum Zitat Zhang, C., Chen, H.: Block boundary value methods for delay differential equations. Appl. Numer. Math. 60, 915–923 (2010) MathSciNetMATH Zhang, C., Chen, H.: Block boundary value methods for delay differential equations. Appl. Numer. Math. 60, 915–923 (2010) MathSciNetMATH
44.
Zurück zum Zitat Zhang, C., Chen, H.: Asymptotic stability of block boundary value methods for delay differential-algebraic equations. Math. Comput. Simul. 81, 100–108 (2010) MathSciNetMATH Zhang, C., Chen, H.: Asymptotic stability of block boundary value methods for delay differential-algebraic equations. Math. Comput. Simul. 81, 100–108 (2010) MathSciNetMATH
45.
Zurück zum Zitat Zhang, C., Chen, H., Wang, L.: Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations. Numer. Linear Alge. Appl. 18, 843–855 (2011) MathSciNetMATH Zhang, C., Chen, H., Wang, L.: Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations. Numer. Linear Alge. Appl. 18, 843–855 (2011) MathSciNetMATH
46.
Zurück zum Zitat Zhang, C., Li, C.: Generalized Störmer-Cowell methods for nonlinear BVPs of second-order delay-integro-differential equations. J. Sci. Comput. 74, 1221–1240 (2018) MathSciNetMATH Zhang, C., Li, C.: Generalized Störmer-Cowell methods for nonlinear BVPs of second-order delay-integro-differential equations. J. Sci. Comput. 74, 1221–1240 (2018) MathSciNetMATH
47.
Zurück zum Zitat Zhou, Y., Zhang, C., Wang, H.: The extended boundary value methods for a class of fractional differential equations with Caputo derivatives, submitted Zhou, Y., Zhang, C., Wang, H.: The extended boundary value methods for a class of fractional differential equations with Caputo derivatives, submitted
48.
Zurück zum Zitat Zhou, Y., Zhang, C.: Convergence and stability of block boundary value methods applied to nonlinear fractional differential equations with Caputo derivatives. Appl. Numer. Math. 135, 367–380 (2019) MathSciNetMATH Zhou, Y., Zhang, C.: Convergence and stability of block boundary value methods applied to nonlinear fractional differential equations with Caputo derivatives. Appl. Numer. Math. 135, 367–380 (2019) MathSciNetMATH
Metadaten
Titel
Preconditioned quasi-compact boundary value methods for space-fractional diffusion equations
verfasst von
Yongtao Zhou
Chengjian Zhang
Luigi Brugnano
Publikationsdatum
10.07.2019
Verlag
Springer US
Erschienen in
Numerical Algorithms / Ausgabe 2/2020
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00773-z

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