A vertex-centred finite volume method for the 3D multi-term time and space fractional Bloch-Torrey equation with fractional Laplacian. (English) Zbl 1503.65206
The authors study anomalous diffusion in biological tissues using a multi-term time-space FBTE (fractional Bloch-Torrey) system on 3D domains. An alternative space-fractional derivative which is the fractional Laplacian is considered. A vertex-centred finite volume method is developed on unstructured tetrahedral meshes. This development provides a setting to approximate the fractional Laplacian via the matrix transfer technique. A Krylov subspace framework is then used to approximate the required matrix function approximations so that the sparsity of \(m(-\Delta)\) ) can be exploited to reduce the computational overhead of the algorithm. An adaptive preconditioner is implemented to accelerate the rate of convergence. Some block matrix functions to deal with the coupled system are introduced. The time-fractional derivatives in a multi-term formulation are discretised by the WSGL (Weighted and shifted Grünwald-Letnikov) formula. Some numerical tests are presented to show the efficiency of the numerical scheme.
Reviewer: Abdallah Bradji (Annaba)
MSC:
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 65F08 | Preconditioners for iterative methods |
| 65F50 | Computational methods for sparse matrices |
| 33E12 | Mittag-Leffler functions and generalizations |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 78A50 | Antennas, waveguides in optics and electromagnetic theory |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
Keywords:
fractional Laplacian; Bloch-Torrey equation; vertex-centred finite volume method; preconditioned Lanczos methodReferences:
| [1] | Taylor, D.; Bushell, M., The spatial mapping of translational diffusion coefficients by the NMR imaging technique, Phys Med Biol, 30, 4, 345 (1985) |
| [2] | Moseley, M.; Kucharczyk, J.; Mintorovitch, J.; Cohen, Y.; Kurhanewicz, J.; Derugin, N., Diffusion-weighted MR imaging of acute stroke: correlation with T2-weighted and magnetic susceptibility-enhanced MR imaging in cats, Am J Neuroradiol, 11, 3, 423-429 (1990) |
| [3] | Le Bihan, D.; Breton, E.; Lallemand, D.; Grenier, P.; Cabanis, E.; Laval-Jeantet, M., MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders, Radiology, 161, 2, 401-407 (1986) |
| [4] | Young, G. S., Advanced MRI of adult brain tumors, Neurol Clin, 25, 4, 947-973 (2007) |
| [5] | Torrey, H. C., Bloch equations with diffusion terms, Phys Rev, 104, 3, 563 (1956) |
| [6] | Le Bihan, D.; Breton, E.; Lallemand, D.; Aubin, M.; Vignaud, J.; Laval-Jeantet, M., Separation of diffusion and perfusion in intravoxel incoherent motion MR imaging, Radiology, 168, 2, 497-505 (1988) |
| [7] | Pfeuffer, J.; Provencher, S. W.; Gruetter, R., Water diffusion in rat brain in vivo as detected at very large b values is multicompartmental, Mag Reson Mater Phys Biol Med, 8, 2, 98-108 (1999) |
| [8] | Jensen, J. H.; Helpern, J. A.; Ramani, A.; Lu, H.; Kaczynski, K., Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging, Magn Reson Med Official J Int Soc Magn Reson Med, 53, 6, 1432-1440 (2005) |
| [9] | Bennett, K. M.; Schmainda, K. M.; Bennett, R.; Rowe, D. B.; Lu, H.; Hyde, J. S., Characterization of continuously distributed cortical water diffusion rates with a stretched-exponential model, Magn Reson Med Official J Int Soc Magn Reson Med, 50, 4, 727-734 (2003) |
| [10] | Magin, R. L.; Abdullah, O.; Baleanu, D.; Zhou, X. J., Anomalous diffusion expressed through fractional order differential operators in the Bloch-torrey equation, J Magn Reson, 190, 2, 255-270 (2008) |
| [11] | Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010 |
| [12] | Zhou, X. J.; Gao, Q.; Abdullah, O.; Magin, R. L., Studies of anomalous diffusion in the human brain using fractional order calculus, Magn Reson Med, 63, 3, 562-569 (2010) |
| [13] | Sun, H.; Sun, Z.-z.; Gao, G.-h., Some high order difference schemes for the space and time fractional Bloch-Torrey equations, Appl Math Comput, 281, 356-380 (2016) · Zbl 1410.65329 |
| [14] | Zhao, Y.; Bu, W.; Zhao, X.; Tang, Y., Galerkin finite element method for two-dimensional space and time fractional Bloch-Torrey equation, J Comput Phys, 350, 117-135 (2017) · Zbl 1380.65294 |
| [15] | Qin, S.; Liu, F.; Turner, I. W.; Yang, Q.; Yu, Q., Modelling anomalous diffusion using fractional Bloch-Torrey equations on approximate irregular domains, Comput Math Appl, 75, 1, 7-21 (2018) · Zbl 1418.65105 |
| [16] | Qin, S.; Liu, F.; Turner, I. W., A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements, Commun Nonlinear Sci Numer Simul, 56, 270-286 (2018) · Zbl 1510.92116 |
| [17] | Dehghan, M.; Abbaszadeh, M., An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch-Torrey equations, Appl Numer Math, 131, 190-206 (2018) · Zbl 1395.65074 |
| [18] | Liu, F.; Feng, L.; Anh, V.; Li, J., Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equations on irregular convex domains, Comput Math Appl, 78, 5, 1637-1650 (2019) · Zbl 1442.65268 |
| [19] | Xu, T.; Liu, F.; Lü, S.; Anh, V. V., Finite difference/finite element method for two-dimensional time-space fractional Bloch-torrey equations with variable coefficients on irregular convex domains, Comput Math Appl, 80, 12, 3173-3192 (2020) · Zbl 1524.65424 |
| [20] | Kwaśnicki, M., Ten equivalent definitions of the fractional Laplace operator, Fract Calc Appl Anal, 20, 1, 7-51 (2017) · Zbl 1375.47038 |
| [21] | Lischke, A.; Pang, G.; Gulian, M.; Song, F.; Glusa, C.; Zheng, X., What is the fractional Laplacian? A comparative review with new results, J Comput Phys, 404, Article 109009 pp. (2020) · Zbl 1453.35179 |
| [22] | Song, R.; Vondraček, Z., Potential theory of subordinate killed Brownian motion in a domain, Probab Theory Related Fields, 125, 4, 578-592 (2003) · Zbl 1022.60078 |
| [23] | D’Elia, M.; Gunzburger, M., The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput Math Appl, 66, 7, 1245-1260 (2013) · Zbl 1345.35128 |
| [24] | Yang, Q.; Turner, I.; Liu, F.; Ilić, M., Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J Sci Comput, 33, 3, 1159-1180 (2011) · Zbl 1229.35315 |
| [25] | Ding, H.; Li, C., Numerical algorithms for the time-Caputo and space-Riesz fractional Bloch-Torrey equations, Numer Methods Partial Differential Equations, 36, 4, 772-799 (2020) · Zbl 07771414 |
| [26] | Bu, W.; Zhao, Y.; Shen, C., Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation, Appl Math Comput, 398, Article 125985 pp. (2021) · Zbl 1508.65125 |
| [27] | Lu, H.; Li, J.; Zhang, M., Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations, Discrete Contin Dyn Syst Ser B, 25, 9, 3357 (2020) · Zbl 1447.78018 |
| [28] | Wang, Y.; Liu, F.; Mei, L.; Anh, V. V., A novel alternating-direction implicit spectral Galerkin method for a multi-term time-space fractional diffusion equation in three dimensions, Numer Algorithms, 86, 1443-1474 (2021) · Zbl 1471.65166 |
| [29] | Yang, Z.; Liu, F.; Nie, Y.; Turner, I., An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains, J Comput Phys, 408, Article 109284 pp. (2020) · Zbl 07505619 |
| [30] | Barth, T.; Herbin, R.; Ohlberger, M., Finite volume methods: foundation and analysis, (Encyclopedia of computational mechanics second edition (2018), Wiley Online Library), 1-60 |
| [31] | Baliga, B.; Patankar, S., A control volume finite-element method for two-dimensional fluid flow and heat transfer, Numer Heat Transfer, 6, 3, 245-261 (1983) · Zbl 0557.76097 |
| [32] | Selmin, V., The node-centred finite volume approach: bridge between finite differences and finite elements, Comput Methods Appl Mech Engrg, 102, 1, 107-138 (1993) · Zbl 0767.76058 |
| [33] | Calgaro, C.; Chane-Kane, E.; Creusé, E.; Goudon, T., L-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios, J Comput Phys, 229, 17, 6027-6046 (2010) · Zbl 1425.76157 |
| [34] | Cordazzo J, Maliska CR, Silva A, Hurtado FS. The negative transmissibility issue when using CVFEM in petroleum reservoir simulation-1. Theory. In: Proceedings of the 10th Brazilian congress of thermal sciences and engineering. 2004. |
| [35] | Ilić, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation, i, Fract Calc Appl Anal, 8, 3, 323-341 (2005) · Zbl 1126.26009 |
| [36] | Lanczos, C., An iteration method for the solution of the eigenvalue problem of linear differential and integral operators (1950), United States Governm. Press Office Los Angeles: United States Governm. Press Office Los Angeles CA · Zbl 0067.33703 |
| [37] | Erhel, J.; Burrage, K.; Pohl, B., Restarted GMRES preconditioned by deflation, J Comput Appl Math, 69, 2, 303-318 (1996) · Zbl 0854.65025 |
| [38] | Baglama, J.; Calvetti, D.; Golub, G. H.; Reichel, L., Adaptively preconditioned GMRES algorithms, SIAM J Sci Comput, 20, 1, 243-269 (1998) · Zbl 0954.65026 |
| [39] | Wang, Z.; Vong, S., Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J Comput Phys, 277, 1-15 (2014) · Zbl 1349.65348 |
| [40] | Ilić, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation (II)-with nonhomogeneous boundary conditions, Fract Calc Appl Anal, 9, 4, 333-349 (2006) · Zbl 1132.35507 |
| [41] | Zienkiewicz, O. C.; Taylor, R. L.; Nithiarasu, P.; Zhu, J., The finite element method. Vol. 3 (1977), McGraw-hill London · Zbl 0435.73072 |
| [42] | Yang, Q.; Moroney, T.; Burrage, K.; Turner, I.; Liu, F., Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions, ANZIAM J, 52, C395-C409 (2010) · Zbl 1390.65093 |
| [43] | Tian, W.; Zhou, H.; Deng, W., A class of second order difference approximations for solving space fractional diffusion equations, Math Comp, 84, 294, 1703-1727 (2015) · Zbl 1318.65058 |
| [44] | Van der Vorst, H. A., An iterative solution method for solving \(f ( A ) x = b\), using Krylov subspace information obtained for the symmetric positive definite matrix a, J Comput Appl Math, 18, 2, 249-263 (1987) · Zbl 0621.65022 |
| [45] | Saad, Y., Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J Numer Anal, 29, 1, 209-228 (1992) · Zbl 0749.65030 |
| [46] | Druskin, V.; Knizhnerman, L., Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic, Numer Linear Algebra Appl, 2, 3, 205-217 (1995) · Zbl 0831.65042 |
| [47] | Stathopoulos, A.; Saad, Y.; Wu, K., Dynamic thick restarting of the Davidson, and the implicitly restarted arnoldi methods, SIAM J Sci Comput, 19, 1, 227-245 (1998) · Zbl 0924.65028 |
| [48] | Ilic, M.; Turner, I., Approximating functions of a large sparse positive definite matrix using a spectral splitting method, ANZIAM J, 46, C472-C487 (2005) · Zbl 1078.65540 |
| [49] | Ilić, M.; Turner, I.; Anh, V., A numerical solution using an adaptively preconditioned lanczos method for a class of linear systems related with the fractional Poisson equation, J Appl Math Stoch Anal, 2008 (2008) · Zbl 1162.65015 |
| [50] | Ben-Israel, A., A note on partitioned matrices and equations, Siam Rev, 11, 2, 247-250 (1969) · Zbl 0175.02402 |
| [51] | Sandev, T.; Tomovski, Ž., Fractional equations and models: theory and applications. Vol. 61 (2019), Springer Nature · Zbl 1473.35002 |
| [52] | Hilfer, R.; Luchko, Y.; Tomovski, Z., Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract Calc Appl Anal, 12, 3, 299-318 (2009) · Zbl 1182.26011 |
| [53] | Prabhakar, T. R., A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math J, 19, 1, 7-15 (1971) · Zbl 0221.45003 |
| [54] | Zeng, F.; Zhang, Z.; Karniadakis, G. E., Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions, Comput Methods Appl Mech Engrg, 327, 478-502 (2017) · Zbl 1439.65081 |
| [55] | Feng, L.; Turner, I.; Perré, P.; Burrage, K., An investigation of nonlinear time-fractional anomalous diffusion models for simulating transport processes in heterogeneous binary media, Commun Nonlinear Sci Numer Simul, 92, Article 105454 pp. (2021) · Zbl 1452.76227 |
| [56] | Zhang, M.; Liu, F.; Turner, I. W.; Anh, V. V.; Feng, L., A finite volume method for the two-dimensional time and space variable-order fractional Bloch-Torrey equation with variable coefficients on irregular domains, Comput Math Appl, 98, 81-98 (2021) · Zbl 1504.65188 |
| [57] | Padhani, A. R.; Liu, G.; Mu-Koh, D.; Chenevert, T. L.; Thoeny, H. C.; Takahara, T., Diffusion-weighted magnetic resonance imaging as a cancer biomarker: consensus and recommendations, Neoplasia, 11, 2, 102-125 (2009) |
| [58] | Hall, M. G.; Barrick, T. R., From diffusion-weighted MRI to anomalous diffusion imaging, Magn Reson Med Official J Int Soc Magn Reson Med, 59, 3, 447-455 (2008) |
| [59] | Feng, Y.; Okamoto, R. J.; Namani, R.; Genin, G. M.; Bayly, P. V., Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter, J Mech Behav Biomed Mater, 23, 117-132 (2013) |
| [60] | Budday, S.; Nay, R.; de Rooij, R.; Steinmann, P.; Wyrobek, T.; Ovaert, T. C., Mechanical properties of gray and white matter brain tissue by indentation, J Mech Behav Biomed Mater, 46, 318-330 (2015) |
| [61] | Ingo, C.; Magin, R. L.; Parrish, T. B., New insights into the fractional order diffusion equation using entropy and kurtosis, Entropy, 16, 11, 5838-5852 (2014) |
| [62] | Szczepankiewicz, F.; van Westen, D.; Englund, E.; Westin, C.-F.; Ståhlberg, F.; Lätt, J., The link between diffusion MRI and tumor heterogeneity: Mapping cell eccentricity and density by diffusional variance decomposition (DIVIDE), Neuroimage, 142, 522-532 (2016) |
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