A Generalized BDF2-θ Finite Element Method for Nonlinear Distributed-Order Time-Fractional Hyperbolic Wave Equations
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摘要: 构造了一种基于带有位移参数θ的广义向后差分公式(广义BDF2-θ)的有限元(FE)方法，用于求解非线性时间分布阶双曲波动方程. 时间方向由广义BDF2-θ近似进一步得到FE全离散格式. 将具有高阶时间导数的模型转化为包括两个低阶方程的耦合系统. 证明了格式的稳定性以及两个函数u和p的最优误差估计结果. 最后，通过数值算例验证了格式的可行性和有效性.Abstract: A finite element (FE) method based on the generalized backward differentiation θ formula (generalized BDF2-θ) was presented to solve nonlinear distributed-order time-fractional hyperbolic wave equations. The temporal direction was approximated with the generalized BDF2-θ to get the FE fully discrete scheme. The proposed model with high-order temporal derivatives was transformed into a coupled system including 2 lower-order equations. The stability of the proposed FE scheme and the optimal error estimates for 2 functions u and p were discussed. Several numerical examples indicate the feasibility and efficiency of the schemes.
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图 1 当θ=0.2, Δα=1/400, Δt=hx=hy=1/8时, 在t=0.5处的数值解uh
Figure 1. The numerical solution uh at t=0.5 with θ=0.2, Δα=1/400, Δt=hx=hy=1/8
图 2 当θ=0.2, Δα=1/400, Δt=hx=hy=1/16时, 在t=0.5处的数值解uh
Figure 2. The numerical solution uh at t=0.5 with θ=0.2, Δα=1/400, Δt=hx=hy=1/16
图 3 当θ=0.2, Δα=1/400, Δt=hx=hy=1/32时, 在t=0.5处的数值解uh
Figure 3. The numerical solution uh at t=0.5 with θ=0.2, Δα=1/400, Δt=hx=hy=1/32
图 4 当θ=0.2, Δα=1/400, Δt=hx=hy=1/64时, 在t=0.5处的数值解uh
Figure 4. The numerical solution uh at t=0.5 with θ=0.2, Δα=1/400, Δt=hx=hy=1/64
图 5 当θ=0.2, Δα=1/400, Δt=hx=hy=1/8时, 在t=0.5处的误差u-uh
Figure 5. The error u-uh at t=0.5 with θ=0.2, Δα=1/400, Δt=hx=hy=1/8
图 6 当θ=0.2, Δα=1/400, Δt=hx=hy=1/16时, 在t=0.5处的误差u-uh
Figure 6. The error u-uh at t=0.5 with θ=0.2, Δα=1/400, Δt=hx=hy=1/16
图 7 当θ=0.2, Δα=1/400, Δt=hx=hy=1/32时, 在t=0.5处的误差u-uh
Figure 7. The error u-uh at t=0.5 with θ=0.2, Δα=1/400, Δt=hx=hy=1/32
表 1 当θ=0.2, Δα=1/400, Δt=hx=hy=1/8, 1/16, 1/32, 1/64时的误差和收敛阶
Table 1. The space-time errors and convergence orders with θ=0.2, Δα=1/400, Δt=hx=hy=1/8, 1/16, 1/32, 1/64
Δt hx=hy ‖u-uh‖ rate T/s 1/8 1/8 5.336 4×10-3 - 0.61 1/16 1/16 1.477 5×10-3 1.852 7 2.54 1/32 1/32 3.784 5×10-4 1.965 0 15.57 1/64 1/64 9.518 8×10-5 1.991 3 134.66 
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表 2 当θ=0.5, Δα=1/400, Δt=hx=hy=1/8, 1/16, 1/32, 1/64时的误差和收敛阶
Table 2. The space-time errors and convergence orders with θ=0.5, Δα=1/400, Δt=hx=hy=1/8, 1/16, 1/32, 1/64
Δt hx=hy ‖u-uh‖ rate T/s 1/8 1/8 1.294 6×10-3 - 1.92 1/16 1/16 3.326 9×10-4 1.960 3 2.57 1/32 1/32 8.461 5×10-5 1.975 2 16.45 1/64 1/64 2.137 7×10-5 1.984 9 130.07
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[1] TAN Z J, ZENG Y H. Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations[J]. Applied Mathematics and Computation, 2024, 466: 128457. doi: 10.1016/j.amc.2023.128457 [2] CAO Y, YIN B L, LIU Y, et al. Crank-Nicolson WSGI difference scheme with finite element method for multi-dimensional time-fractional wave problem[J]. Computational and Applied Mathematics, 2018, 37(4): 5126-5145. doi: 10.1007/s40314-018-0626-2 [3] CHEN Y P, GU Q L, LI Q F, et al. A two-grid finite element approximation for nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations[J]. Journal of Computational Mathematics, 2022, 40(6): 936-954. doi: 10.4208/jcm.2104-m2021-0332 [4] DING H F, LI C P. A high-order algorithm for time-caputo-tempered partial differential equation with riesz derivatives in two spatial dimensions[J]. Journal of Scientific Computing, 2019, 80(1): 81-109. doi: 10.1007/s10915-019-00930-5 [5] LI L M, XU D, LUO M. Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation[J]. Journal of Computational Physics, 2013, 255: 471-485. doi: 10.1016/j.jcp.2013.08.031 [6] REN J C, SUN Z Z. Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions[J]. Journal of Scientific Computing, 2013, 56(2): 381-408. doi: 10.1007/s10915-012-9681-9 [7] CHEN M H, DENG W H. A second-order accurate numerical method for the space-time tempered fractional diffusion-wave equation[J]. Applied Mathematics Letters, 2017, 68: 87-93. doi: 10.1016/j.aml.2016.12.010 [8] WANG Z B, VONG S. Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation[J]. Journal of Computational Physics, 2014, 277: 1-15. doi: 10.1016/j.jcp.2014.08.012 [9] YU B, JIANG X Y, WANG C. Numerical algorithms to estimate relaxation parameters and Caputo fractional derivative for a fractional thermal wave model in spherical composite medium[J]. Applied Mathematics and Computation, 2016, 274: 106-118. doi: 10.1016/j.amc.2015.10.081 [10] ZENG F H. Second-order stable finite difference schemes for the time-fractional diffusion-wave equation[J]. Journal of Scientific Computing, 2015, 65(1): 411-430. doi: 10.1007/s10915-014-9966-2 [11] DU R L, SUN Z Z, WANG H. Temporal second-order finite difference schemes for variable-order time-fractional wave equations[J]. SIAM Journal on Numerical Analysis, 2022, 60(1): 104-132. doi: 10.1137/19M1301230 [12] CAO F F, ZHAO Y M, WANG F L, et al. Nonconforming mixed FEM analysis for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with time-space coupled derivative[J]. Advances in Applied Mathematics and Mechanics, 2023, 15(2): 322-358. doi: 10.4208/aamm.OA-2021-0263 [13] 吴迪, 李小林. 时间分数阶扩散波方程的无单元Galerkin法分析[J]. 应用数学和力学, 2022, 43(2): 215-223.WU Di, LI Xiaolin. An element-free Galerkin method for time-fractional diffusion-wave equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 215-223. (in Chinese) [14] HEYDARI M H, HOOSHMANDASL M R, GHAINI F M M, et al. Wavelets method for the time fractional diffusion-wave equation[J]. Physics Letters A, 2015, 379(3): 71-76. doi: 10.1016/j.physleta.2014.11.012 [15] 刘家惠, 邵林馨, 黄健飞. 带Caputo导数的变分数阶随机微分方程的Euler-Maruyama方法[J]. 应用数学和力学, 2023, 44(6): 731-743. doi: 10.21656/1000-0887.430250LIU Jiahui, SHAO Linxin, HUANG Jianfei. An Euler-Maruyama method for variable fractional stochastic differential equations with caputo derivatives[J]. Applied Mathematics and Mechanics, 2023, 44(6): 731-743. (in Chinese) doi: 10.21656/1000-0887.430250 [16] 汪精英, 翟术英. 分数阶Cahn-Hilliard方程的高效数值算法[J]. 应用数学和力学, 2021, 42(8): 832-840. doi: 10.21656/1000-0887.420008WANG Jingying, ZHAI Shuying. An efficient numerical algorithm for fractional Cahn-Hilliard equations[J]. Applied Mathematics and Mechanics, 2021, 42(8): 832-840. (in Chinese) doi: 10.21656/1000-0887.420008 [17] CHECHKIN A V, GORENFLO R, SOKOLOV I M. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations[J]. Physical Review E, 2002, 66(4): 046129. doi: 10.1103/PhysRevE.66.046129 [18] NIU Y X, LIU Y, LI H, et al. Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media[J]. Mathematics and Computers in Simulation, 2023, 203: 387-407. doi: 10.1016/j.matcom.2022.07.001 [19] WEI L L, LIU L J, SUN H X. Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order[J]. Journal of Applied Mathematics and Computing, 2019, 59(1): 323-341. http://www.zhangqiaokeyan.com/journal-foreign-detail/0704025378140.html [20] DIETHELM K, FORD N J. Numerical analysis for distributed-order differential equations[J]. Journal of Computational and Applied Mathematics, 2009, 225(1): 96-104. doi: 10.1016/j.cam.2008.07.018 [21] RAN M, ZHANG C J. New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order[J]. Applied Numerical Mathematics, 2018, 129: 58-70. doi: 10.1016/j.apnum.2018.03.005 [22] BU W P, XIAO A G, ZENG W. Finite difference/finite element methods for distributed-order time fractional diffusion equations[J]. Journal of Scientific Computing, 2017, 72(1): 422-441. doi: 10.1007/s10915-017-0360-8 [23] YIN B L, LIU Y, LI H, et al. Approximation methods for the distributed order calculus using the convolution quadrature[J]. Discrete & Continuous Dynamical Systems B, 2021, 26(3): 1447-1468. [24] WEN C, LIU Y, YIN B L, et al. Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model[J]. Numerical Algorithms, 2021, 88(2): 523-553. doi: 10.1007/s11075-020-01048-8 [25] ZHANG H, LIU F W, JIANG X Y, et al. A Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional Riesz space distributed-order advection-diffusion equation[J]. Computers & Mathematics With Applications, 2018, 76(10): 2460-2476. http://www.sciencedirect.com/science/article/pii/S0898122118304681 [26] JIAN H Y, HUANG T Z, GU X M, et al. Fast second-order implicit difference schemes for time distributed-order andRiesz space fractional diffusion-wave equations[J]. Computers & Mathematics With Applications, 2021, 94: 136-154. [27] ATANACKOVIC T M, PILIPOVIC S, ZORICA D. Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod[J]. International Journal of Engineering Science, 2011, 49(2): 175-190. doi: 10.1016/j.ijengsci.2010.11.004 [28] GORENFLO R, LUCHKO Y, STOJANOVIĆ M. Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density[J]. Fractional Calculus and Applied Analysis, 2013, 16(2): 297-316. doi: 10.2478/s13540-013-0019-6 [29] YE H, LIU F W, ANH V. Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains[J]. Journal of Computational Physics, 2015, 298: 652-660. doi: 10.1016/j.jcp.2015.06.025 [30] GAO G H, SUN Z Z. Two alternating direction implicit difference schemes for solving the two-dimensional time distributed-order wave equations[J]. Journal of Scientific Computing, 2016, 69(2): 506-531. doi: 10.1007/s10915-016-0208-7 [31] TOMOVSKI Ž, SANDEV T. Distributed-order wave equations with composite time fractional derivative[J]. International Journal of Computer Mathematics, 2018, 95(6/7): 1100-1113. [32] HENDY A S, DE STAELEN R H, PIMENOV V G. A semi-linear delayed diffusion-wave system with distributed order in time[J]. Numerical Algorithms, 2018, 77(3): 885-903. doi: 10.1007/s11075-017-0344-7 [33] DEHGHAN M, ABBASZADEH M. A Legendre spectral element method (SEM) based on the modified bases for solving neutral delay distributed-order fractional damped diffusion-wave equation[J]. Mathematical Methods in the Applied Sciences, 2018, 41(9): 3476-3494. doi: 10.1002/mma.4839 [34] LI X L, RUI H X. A block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation[J]. Applied Numerical Mathematics, 2018, 131: 123-139. doi: 10.1016/j.apnum.2018.04.013 [35] JANNO J. Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data[J]. Fractional Calculus and Applied Analysis, 2020, 23(6): 1678-1701. doi: 10.1515/fca-2020-0083 [36] ENGSTRÖM C, GIANI S, GRUBIŠIĆ L. Numerical solution of distributed-order time-fractional diffusion-wave equations using Laplace transforms[J]. Journal of Computational and Applied Mathematics, 2023, 425: 115035. doi: 10.1016/j.cam.2022.115035 [37] YIN B L, LIU Y, LI H. A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations[J]. Applied Mathematics and Computation, 2020, 368: 124799. doi: 10.1016/j.amc.2019.124799 [38] LIU Y, DU Y W, LI H, et al. Some second-order θ schemes combined with finite element method for nonlinear fractional cable equation[J]. Numerical Algorithms, 2019, 80(2): 533-555. doi: 10.1007/s11075-018-0496-0 [39] GHURAIBAWI A A, MARASI H R, DERAKHSHAN M H, et al. An efficient numerical method for the time-fractional distributed order nonlinear Klein-Gordon equation with shifted fractional Gegenbauer multi-wavelets method[J]. Physica Scripta, 2023, 98(8): 084001. doi: 10.1088/1402-4896/accedb [40] WANGJ F, YIN B L, LIU Y, et al. Mixed finite element algorithm for a nonlinear time fractional wave model[J]. Mathematics and Computers in Simulation, 2021, 188: 60-76. doi: 10.1016/j.matcom.2021.03.038 [41] DURÁN R G, LOMBARDI A L. Finite element approximation of convection diffusion problems using graded meshes[J]. Applied Numerical Mathematics, 2006, 56(10/11): 1314-1325. [42] STYNES M, O'RIORDAN E, GRACIA J L. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation[J]. SIAM Journal on Numerical Analysis, 2017, 55(2): 1057-1079. doi: 10.1137/16M1082329 [43] CHEN H B, XU D, ZHOU J. A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel[J]. Journal of Computational and Applied Mathematics, 2019, 356: 152-163. doi: 10.1016/j.cam.2019.01.031 [44] YANG Z, ZENG F H. A corrected L1 method for a time-fractional subdiffusion equation[J]. Journal of Scientific Computing, 2023, 95(3): 85. doi: 10.1007/s10915-023-02204-7 [45] YIN B L, LIU Y, LI H, et al. Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes[J]. BIT Numerical Mathematics, 2022, 62(2): 631-666. doi: 10.1007/s10543-021-00890-z -
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