×

zbMATH — the first resource for mathematics

A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation. (English) Zbl 07249433
Summary: This paper introduces a high-order numerical procedure to solve the two-dimensional distributed-order Riesz space-fractional diffusion equation. In the proposed technique, first, a second-order numerical integration rule is employed to estimate the integral of the distributed-order Riesz space-fractional derivative. Then, the time derivative is discretized by a second-order difference scheme. Finally, the spatial direction is approximated by a difference formulation with fourth-order accuracy. The stability of the semi-discrete scheme is analyzed. We conclude that the difference between two consecutive time steps i.e. \(U_{i, j}^n - U_{i, j}^{n - 1}\) is nearly zero when \(n \to \infty\). So, a suitable term is added to the main difference scheme as by adding this term we could derive the main ADI scheme. Furthermore, to reduce the used CPU time, we combine the fourth-order ADI formulation with the proper orthogonal decomposition method and then we gain a POD based reduced-order compact ADI finite difference plane. In the next, the convergence order of the fully discrete formulation has been investigated. The numerical results show the efficiency of new technique. It must be noted that the finite difference method is an effective and robust numerical technique for solving nonlinear equations that the ADI approach can be combined with it to improve the numerical simulations.
MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65D30 Numerical integration
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
35K57 Reaction-diffusion equations
Software:
FODE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abbaszadeh, M., Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation, Appl. Math. Lett., 88, 179-185 (2019) · Zbl 1410.65351
[2] Abbaszadeh, M.; Dehghan, M., Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method, Appl. Numer. Math., 145, 488-506 (2019) · Zbl 1428.65073
[3] Abbaszadeh, M.; Dehghan, M., A finite-difference procedure to solve weakly singular integro partial differential equation with space-time fractional derivatives, Eng. Comput. (2020)
[4] Abbaszadeh, M.; Dehghan, M.; Zhou, Y., Crank-Nicolson/Galerkin spectral method for solving two-dimensional time-space distributed-order weakly singular integro-partial differential equation, J. Comput. Appl. Math., 374, Article 112739 pp. (2020) · Zbl 1435.65170
[5] Alikhanov, A. A., A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280, 424-438 (2015) · Zbl 1349.65261
[6] Alikhanov, A. A., Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation, Appl. Math. Comput., 268, 12-22 (2015) · Zbl 1410.65294
[7] Atangana, A., Derivative with two fractional orders: a new avenue of investigation toward revolution in fractional calculus, Eur. Phys. J. Plus, 131, 10, 373 (2016)
[8] Atangana, A.; Jain, S., A new numerical approximation of the fractal ordinary differential equation, Eur. Phys. J. Plus, 133, 2, 37 (2018)
[9] Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 1, 539-575 (1993)
[10] Bhrawy, A. H.; Zaky, M. A., A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys., 281, 876-895 (2015) · Zbl 1352.65386
[11] Bhrawy, A. H.; Zaky, M. A., An improved collocation method for multi-dimensional space-time variable-order fractional Schrodinger equations, Appl. Numer. Math., 111, 197-218 (2017) · Zbl 1353.65106
[12] Bu, W.; Tang, Y.; Yang, J., Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276, 26-38 (2014) · Zbl 1349.65441
[13] Bu, W.; Liu, X.; Tang, Y.; Yang, J., Finite element multigrid method for multi-term time fractional advection diffusion equations, Int. J. Model. Simul. Sci. Comput., 26, 1, Article 1540001 pp. (2015)
[14] Bu, W.; Tang, Y.; Wu, Y.; Yang, J., Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations, J. Comput. Phys., 293, 264-279 (2015) · Zbl 1349.65440
[15] Bu, W.; Tang, Y.; Wu, Y.; Yang, J., Crank-Nicolson ADI Galerkin finite element method for two-dimensional fractional FitzHugh-Nagumo monodomain model, Appl. Math. Comput., 257, 355-364 (2015) · Zbl 1339.65170
[16] Chaturantabut, S., Dimension reduction for unsteady nonlinear partial differential equations via empirical interpolation methods, Proquest (2009)
[17] Chaturantabut, S.; Sorensen, D. C., A state space error estimate for POD-DEIM nonlinear model reduction, SIAM J. Numer. Anal., 50, 1, 46-63 (2012) · Zbl 1237.93035
[18] Chen, X.; Di, Y.; Duan, J.; Li, D. F., Linearized compact ADI schemes for nonlinear time-fractional Schrödinger equations, Appl. Math. Lett., 84, 160-167 (2018) · Zbl 06892656
[19] Cheng, X.; Duan, J.; Li, D. F., A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations, Appl. Math. Comput., 346, 452-464 (2019) · Zbl 1429.65216
[20] Dehghan, M.; Abbaszadeh, M., A finite element method for the numerical solution of Rayleigh- Stokes problem for a heated generalized second grade fluid with fractional derivatives, Eng. Comput., 33, 587-605 (2017)
[21] 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
[22] Dehghan, M.; Abbaszadeh, M., A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation, Comput. Math. Appl., 75, 8, 2903-2914 (2018) · Zbl 1415.65224
[23] Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: meshless interpolating element free Galerkin (IEFG) and finite element methods, Eng. Anal. Bound. Elem., 64, 205-221 (2016) · Zbl 1403.65068
[24] Deng, W., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47, 1, 204-226 (2008) · Zbl 1416.65344
[25] Du, J.; Navon, I. M.; Steward, J. L.; Alekseev, A. K.; Luo, Z., Reduced-order modeling based on pod of a parabolized Navier-Stokes equation model I: forward model, Int. J. Numer. Methods Fluids, 69, 710-730 (2012)
[26] Du, J.; Navon, I. M.; Zhu, J.; Fang, F.; Alekseev, A. K., Reduced order modeling based on pod of a parabolized Navier-Stokes equations model II: trust region POD 4D var data assimilation, Comput. Math. Appl., 65, 380-394 (2013) · Zbl 1319.76030
[27] Ervin, V. J.; Roop, J. P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differ. Equ., 22, 558-576 (2006) · Zbl 1095.65118
[28] Fairweather, G.; Yang, X.; Xu, D.; Zhang, H., An ADI Crank-Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation, J. Sci. Comput., 65, 1217-1239 (2015) · Zbl 1328.65216
[29] Fan, W.; Liu, F., A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain, Appl. Math. Lett., 77, 114-121 (2018) · Zbl 1380.65260
[30] Fan, W.; Liu, F.; Jiang, X.; Turner, I., A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain, Fract. Calc. Appl. Anal., 20, 2, 352-383 (2017) · Zbl 1364.65162
[31] Fang, F.; Pain, C. C.; Navon, I. M.; Gorman, G. J.; Piggott, M. D.; Allison, P. A.; Farrell, P. E.; Goddard, A. J.H., A POD reduced order unstructured mesh ocean modelling method for moderate Reynolds number flows, Ocean Model., 28, 127-136 (2009)
[32] Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I.; Gu, Y. T., Finite element method for space-time fractional diffusion equation, Numer. Algorithms, 72, 749-767 (2016) · Zbl 1343.65122
[33] Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I.; Anh, V.; Li, J., A fast second-order accurate method for a two-sided space-fractional diffusion equation with variable coefficients, Comput. Math. Appl., 73, 1155-1171 (2017) · Zbl 1412.65072
[34] Gao, G. H.; Sun, Z. Z., Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations, J. Sci. Comput., 66, 1281-1312 (2016) · Zbl 1373.65055
[35] Guan, Q.; Gunzburger, M., θ schemes for finite element discretization of the space-time fractional diffusion equations, J. Comput. Appl. Math., 288, 264-273 (2015) · Zbl 1320.65141
[36] Hamid, M.; Usman, M.; Zubair, T.; Mohyud-Din, S. T., Comparison of Lagrange multipliers for telegraph equations, Ain Shams Eng. J., 9, 2323-2328 (2018)
[37] Hamid, M.; Usman, M.; Zubair, T.; Haq, R. U.; Wang, W., Innovative operational matrices based computational scheme for fractional diffusion problems with the Riesz derivative, Eur. Phys. J. Plus, 134, 484 (2019)
[38] Hamid, M.; Zubair, T.; Usma, M.; Haq, R. U., Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel, AIMS Math., 4, 5, 1416-1429 (2019)
[39] Hamid, M.; Usman, M.; Haq, R. U.; Wang, W., A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model, Physica A, 551, Article 124227 pp. (2020)
[40] Hao, Z. P.; Sun, Z. Z.; Cao, W. R., A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys., 281, 787-805 (2015) · Zbl 1352.65238
[41] Jia, J.; Wang, H., A fast finite volume method for conservative space-fractional diffusion equations in convex domains, J. Comput. Phys., 310, 63-84 (2016) · Zbl 1349.65562
[42] Jia, J.; Wang, H., A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains, Comput. Math. Appl., 75, 2031-2043 (2018) · Zbl 1409.65054
[43] Jin, B.; Lazarov, R.; Pasciak, J.; Zhou, Z., Error analysis of a finite element method for the space-fractional parabolic equation, SIAM J. Numer. Anal., 52, 5, 2272-2294 (2013) · Zbl 1310.65126
[44] Kerschen, G.; Golinval, J.; Vakakis, A. F.; Bergman, L. A., The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview, Nonlinear Dyn., 41, 1-3, 147-169 (2005) · Zbl 1103.70011
[45] Li, J.; Liu, F.; Feng, L.; Turner, I., A novel finite volume method for the Riesz space distributed-order diffusion equation, Comput. Math. Appl., 46, 536-553 (2017) · Zbl 1443.65162
[46] Li, L.; Xu, D.; Luo, Man, Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation, J. Comput. Phys., 255, 471-485 (2013) · Zbl 1349.65456
[47] Li, M.; Huang, C., ADI Galerkin FEMs for the 2D nonlinear time-space fractional diffusion-wave equation, Int. J. Model. Simul. Sci. Comput., 8, 3, Article 1750025 pp. (2017)
[48] Li, M.; Zhao, Y. L., A fast energy conserving finite element method for the nonlinear fractional Schrodinger equation with wave operator, Appl. Math. Comput., 338, 758-773 (2018) · Zbl 1427.65253
[49] Li, M.; Gu, X. M.; Huang, C.; Fei, M.; Zhang, G., A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrodinger equations, J. Comput. Phys., 358, 256-282 (2018) · Zbl 1382.65320
[50] Li, M.; Shi, D.; Wang, J.; Ming, W., Unconditional superconvergence analysis of the conservative linearized Galerkin FEMs for nonlinear Klein-Gordon-Schrodinger equation, Appl. Numer. Math., 142, 47-63 (2019) · Zbl 07076640
[51] Li, M.; Zhao, J.; Huang, C.; Chen, S., Nonconforming virtual element method for the time fractional reaction-subdiffusion equation with non-smooth data, J. Sci. Comput., 81, 3, 1823-1859 (2019) · Zbl 1440.65143
[52] Li, M.; Shi, D.; Pei, L., Convergence and superconvergence analysis of finite element methods for the time fractional diffusion equation, Appl. Numer. Math., 151, 141-160 (2020) · Zbl 1435.65127
[53] Li, M.; Shi, D.; Wang, J., Unconditional superconvergence analysis of a linearized Crank-Nicolson Galerkin FEM for generalized Ginzburg-Landau equation, Comput. Math. Appl., 79, 8, 2411-2425 (2020) · Zbl 1437.65197
[54] Lian, Y.; Ying, Y.; Tang, S.; Lin, S.; Wagner, G. J.; Liu, W. K., A Petrov-Galerkin finite element method for the fractional advection-diffusion equation, Comput. Methods Appl. Mech. Eng., 309, 388-410 (2016) · Zbl 1439.65090
[55] Liu, F.; Zhuang, P.; Turner, I.; Anh, V.; Burrage, K., A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain, J. Comput. Phys., 293, 252-263 (2015) · Zbl 1349.65316
[56] Liu, Y.; Fang, Z.; Li, H.; He, S., A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput., 243, 703-717 (2014) · Zbl 1336.65166
[57] Liu, Y.; Du, Y.; Li, H.; Li, J.; He, S., A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative, Comput. Math. Appl., 70, 2474-2492 (2015)
[58] Luo, Z. D.; Chen, J.; Zhu, J.; Wang, R.; Navon, I. M., An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model, Int. J. Numer. Methods Fluids, 55, 143-161 (2007) · Zbl 1205.86007
[59] Luo, Z. D.; Chen, J.; Navon, I. M.; Yang, X., Mixed finite element formulation and error estimate based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations, SIAM J. Numer. Anal., 47, 1-19 (2008)
[60] Macias-Diaz, J. E., Numerical study of the process of nonlinear supratransmission in Riesz space-fractional sine-Gordon equations, Commun. Nonlinear Sci. Numer. Simul., 46, 89-102 (2017)
[61] Pang, H. K.; Sun, H. W., Fourth-order finite difference schemes for time-space fractional sub-diffusion equations, Comput. Math. Appl., 71, 1287-1302 (2016)
[62] Pindza, E.; Owolabi, K. M., Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simul., 40, 112-128 (2016)
[63] Ravindran, S., Reduced-order adaptive controllers for fluid flows using POD, J. Sci. Comput., 15, 4, 457-478 (2000) · Zbl 1048.76016
[64] Ravindran, S. S., A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Int. J. Numer. Methods Fluids, 34, 5, 425-448 (2000) · Zbl 1005.76020
[65] Roop, J. P., Variational solution of the fractional advection dispersion equation (2004), Clemson University, PhD thesis
[66] Song, J.; Yu, Q.; Liu, F.; Turner, I., A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation, Numer. Algorithms, 66, 911-932 (2014) · Zbl 1408.65057
[67] 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
[68] Sun, Z. Z.; Wu, X. N., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 193-209 (2006) · Zbl 1094.65083
[69] Tang, T., A finite difference scheme for a partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11, 309-319 (1993) · Zbl 0768.65093
[70] Tang, T.; Yu, H.; Zhou, T., On energy dissipation theory and numerical stability for time-fractional phase-field equations, SIAM J. Sci. Comput., 41, 6, A3757-A3778 (2019) · Zbl 1435.65146
[71] Tang, T.; Wang, L. L.; Yuan, H.; Zhou, T., Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains, SIAM J. Sci. Comput., 42, 2, A585-A611 (2020) · Zbl 1447.65161
[72] Tian, W.; Zhou, H.; Deng, W., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput., 84, 294, 1703-1727 (2015) · Zbl 1318.65058
[73] Usman, M.; Hamid, M.; Khalid, M. S.U.; Ul Haq, R.; Liu, M., A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics, Numer. Methods Partial Differ. Equ. (2020), in press
[74] Usman, M.; Hamid, M.; Zubair, T.; Haq, R. U.; Wang, W.; Liu, M. B., Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials, Appl. Math. Comput., 372, Article 124985 pp. (2020) · Zbl 1433.65152
[75] Vong, S.; Wang, Z., A compact difference scheme for a two dimensional fractional Klein-Gordon equation with Neumann boundary conditions, J. Comput. Phys., 274, 268-282 (2014) · Zbl 1352.65273
[76] Vong, S.; Lyu, P.; Chen, X.; Lei, S., High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Lliouville derivatives, Numer. Algorithms, 72, 195-210 (2016) · Zbl 1382.65259
[77] Wang, J.; Liu, T.; Li, H.; Liu, Y.; He, S., Second-order approximation scheme combined with \(H^1\)-Galerkin MFE method for nonlinear time fractional convection-diffusion equation, Comput. Math. Appl., 73, 1182-1196 (2017) · Zbl 1412.65157
[78] Wang, Y. M., A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection-diffusion equations, Calcolo, 54, 733-768 (2017) · Zbl 1422.65190
[79] Wang, Y. M.; Wang, T., Error analysis of a high-order compact ADI method for two-dimensional fractional convection-subdiffusion equations, Calcolo, 53, 301-330 (2016) · Zbl 1359.65173
[80] Wang, Y. M.; Wang, T., A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions, Comput. Math. Appl., 75, 721-739 (2018) · Zbl 1409.65058
[81] 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
[82] Wei, L.; Dai, H.; Zhang, D.; Si, Z., Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo, 51, 175-192 (2014) · Zbl 1311.35331
[83] Weng, Z.; Zhai, S.; Feng, X., A Fourier spectral method for fractional-in-space Cahn-Hilliard equation, Appl. Math. Model., 42, 462-477 (2017) · Zbl 1443.65255
[84] Xiao, D.; Fang, F.; Du, J.; Pain, C. C.; Navon, I. M.; Buchan, A. G.; ElSheikh, A. H.; Datum, G. H., Non-linear Petrov-Galerkin methods for reduced order modelling of the Navier-Stokes equations using a mixed finite element pair, Comput. Methods Appl. Mech. Eng., 255, 147-157 (2013) · Zbl 1297.76107
[85] Xiao, D.; Fang, F.; Buchan, A. G.; Pain, C. C.; Navon, I. M.; Du, J.; Hu, G., Non-linear model reduction for the Navier-Stokes equations using residual DEIM method, J. Comput. Phys., 263, 1-18 (2014) · Zbl 1349.76288
[86] Xiao, D.; Fang, F.; Pain, C. C.; Hu, G., Non-intrusive reduced-order modelling of the Navier-Stokes equations based on RBF interpolation, Int. J. Numer. Methods Fluids, 79, 580-595 (2015)
[87] Xiao, D.; Lin, Z.; Fang, F.; Pain, C. C.; Navon, I. M.; Salinas, P.; Muggeridge, A., Non-intrusive reduced-order modeling for multiphase porous media flows using Smolyak sparse grids, Int. J. Numer. Methods Fluids, 83, 2, 205-219 (2017)
[88] Xiao, D.; Yang, P.; Fang, F.; Xiang, J.; Pain, C. C.; Navon, I. M.; Chen, M., A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting, J. Comput. Phys., 330, 221-244 (2017) · Zbl 1378.74067
[89] Xiao, D.; Fang, F.; Pain, C. C.; Navon, I. M.; Salinasa, P., Non-intrusive model reduction for a 3D unstructured mesh control volume finite element reservoir model and its application to fluvial channels, Int. J. Oil Gas, Coal Technol., 19, 316-339 (2018)
[90] Xiao, D.; Fang, F.; Zheng, J.; Pain, C. C.; Navon, I. M., Machine learning-based rapid response tools for regional air pollution modelling, Atmos. Environ., 199, 463-473 (2019)
[91] Yang, Y.; Ma, H., The Legendre Galerkin-Chebyshev collocation method for space fractional Burgers-like equations, Numer. Math., Theory Methods Appl., 11, 338-353 (2018) · Zbl 1424.65197
[92] Yang, Z.; Yuan, Z.; Nie, Y.; Wang, J.; Zhu, X.; Liu, F., Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains, J. Comput. Phys., 330, 863-883 (2017) · Zbl 1378.35330
[93] Yu, Y.; Deng, W.; Wu, Y.; Wu, J., Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations, Appl. Numer. Math., 112, 126-145 (2017) · Zbl 1354.65174
[94] Zaky, M. A., Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems, Appl. Numer. Math., 145, 429-457 (2019) · Zbl 1427.34021
[95] Zaky, M. A., An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions, Appl. Numer. Math., 154, 205-222 (2020) · Zbl 1442.65464
[96] Zaky, M. A.; Hendy, A. S.; Macias-Diaz, J. E., Semi-implicit Galerkin-Legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions, J. Sci. Comput., 82, 1-27 (2020) · Zbl 1433.65247
[97] Zayernouri, M.; Karniadakis, G. E., Exponentially accurate spectral and spectral element methods for fractional ODEs, J. Comput. Phys., 257, 460-480 (2014) · Zbl 1349.65257
[98] Zayernouri, M.; Karniadakis, G. E., Discontinuous spectral element methods for time- and space-fractional advection equations, SIAM J. Sci. Comput., 36, B684-B707 (2014) · Zbl 1304.35757
[99] Zhang, H.; Liu, F.; Jiang, X.; Zeng, F.; Turner, I., A Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional Riesz space distributed-order advection-diffusion equation, Comput. Math. Appl., 76, 2460-2476 (2018)
[100] Zhang, P.; Zhang, X. H.; Xiang, H.; Song, L., A fast and stabilized meshless method for the convection-dominated convection-diffusion problems, Numer. Heat Transf., Part A, Appl., 70, 4, 420-431 (2016)
[101] Zhang, X.; Xiang, H., A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems, Int. J. Heat Mass Transf., 84, 729-739 (2015)
[102] Zhang, Y.; Sun, Z. Z., Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comput., 59, 104-128 (2014) · Zbl 1304.65208
[103] Zhao, X.; Sun, Z. Z.; Hao, Z. P., A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation, SIAM J. Sci. Comput., 36, A2865-A2886 (2014) · Zbl 1328.65187
[104] Zhao, Y.; Bu, W.; Huang, J.; Liu, D. Y.; Tang, Y., Finite element method for two-dimensional space-fractional advection-dispersion equations, Appl. Math. Comput., 257, 553-565 (2015) · Zbl 1339.65185
[105] Zhao, Z.; Li, C. P., Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219, 2975-2988 (2012) · Zbl 1309.65101
[106] Zhuang, P.; Liu, F.; Turner, I.; Gu, Y. T., Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation, Appl. Math. Model., 38, 3860-3870 (2014) · Zbl 1429.65233
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.