A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative
In this paper, an effective numerical method for solving the variable-order(VO) fractional reaction diffusion equation with the Caputo fractional derivative is constructed and analyzed. Based on the generalized alternating numerical flux, we get a fully discrete local discontinuous Galerkin scheme for the problem. From a practical standpoint, the generalized alternating numerical flux, which is distinct from the purely alternating numerical flux, has a more extensive scope. For 0<α(t)<1, we prove that the method is unconditionally stable and the errors attain (k+1)-th order of accuracy for piecewise Pk polynomials. Finally, some numerical experiments are performed to show the effectiveness and verify the accuracy of the method.
Citation: Lijie Liu, Xiaojing Wei, Leilei Wei. A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative[J]. Electronic Research Archive, 2023, 31(9): 5701-5715. doi: 10.3934/era.2023289
Related Papers:
[1]
Leilei Wei, Xiaojing Wei, Bo Tang .
Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation. Electronic Research Archive, 2022, 30(4): 1263-1281.
doi: 10.3934/era.2022066
[2]
Chang Hou, Hu Chen .
Stability and pointwise-in-time convergence analysis of a finite difference scheme for a 2D nonlinear multi-term subdiffusion equation. Electronic Research Archive, 2025, 33(3): 1476-1489.
doi: 10.3934/era.2025069
[3]
Li Tian, Ziqiang Wang, Junying Cao .
A high-order numerical scheme for right Caputo fractional differential equations with uniform accuracy. Electronic Research Archive, 2022, 30(10): 3825-3854.
doi: 10.3934/era.2022195
[4]
Akeel A. AL-saedi, Jalil Rashidinia .
Application of the B-spline Galerkin approach for approximating the time-fractional Burger's equation. Electronic Research Archive, 2023, 31(7): 4248-4265.
doi: 10.3934/era.2023216
[5]
Xingyang Ye, Xiaoyue Liu, Tong Lyu, Chunxiu Liu .
$ \alpha $-robust error analysis of two nonuniform schemes for Caputo-Hadamard fractional reaction sub-diffusion problems. Electronic Research Archive, 2025, 33(1): 353-380.
doi: 10.3934/era.2025018
[6]
Jun Pan, Yuelong Tang .
Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations. Electronic Research Archive, 2023, 31(12): 7207-7223.
doi: 10.3934/era.2023365
[7]
Yining Yang, Yang Liu, Cao Wen, Hong Li, Jinfeng Wang .
Efficient time second-order SCQ formula combined with a mixed element method for a nonlinear time fractional wave model. Electronic Research Archive, 2022, 30(2): 440-458.
doi: 10.3934/era.2022023
[8]
Victor Ginting .
An adjoint-based a posteriori analysis of numerical approximation of Richards equation. Electronic Research Archive, 2021, 29(5): 3405-3427.
doi: 10.3934/era.2021045
[9]
Wenjing An, Xingdong Zhang .
An implicit fully discrete compact finite difference scheme for time fractional diffusion-wave equation. Electronic Research Archive, 2024, 32(1): 354-369.
doi: 10.3934/era.2024017
[10]
Janarthanan Ramadoss, Asma Alharbi, Karthikeyan Rajagopal, Salah Boulaaras .
A fractional-order discrete memristor neuron model: Nodal and network dynamics. Electronic Research Archive, 2022, 30(11): 3977-3992.
doi: 10.3934/era.2022202
Abstract
In this paper, an effective numerical method for solving the variable-order(VO) fractional reaction diffusion equation with the Caputo fractional derivative is constructed and analyzed. Based on the generalized alternating numerical flux, we get a fully discrete local discontinuous Galerkin scheme for the problem. From a practical standpoint, the generalized alternating numerical flux, which is distinct from the purely alternating numerical flux, has a more extensive scope. For 0<α(t)<1, we prove that the method is unconditionally stable and the errors attain (k+1)-th order of accuracy for piecewise Pk polynomials. Finally, some numerical experiments are performed to show the effectiveness and verify the accuracy of the method.
1.
Introduction
Fractional order partial differential equations are a generalization of classical partial differential equations [1,2,3]. The theory of fractional order calculus has a wide range of applications in mathematical physics equations, electrochemical processes, mechanics, anomalous diffusion, processing of signals and finance [4,5,6,7,8]. Since the analytical solutions of many fractional order differential equations cannot be solved exactly [9], numerical methods for fractional order differential equations have attracted a great deal of attention from an increasing number of scholars. Many researchers in domestic and foreign countries have solved many different types of fractional order partial differential equations by different methods. There are spectral methods [10,11,12,13,14,15], finite difference method [16,17,18,19], finite element method [20,21,22,23,24,25], local discontinuous Galerkin methods (LDG) [26,27,28,29,30]. etc.
In recent years, many scholars have studied the Caputo-type reaction diffusion equation. Imran and Shah etal [31] investigated unsteady time fractional natural convection flow of incompressible viscous fluid, and obtained some exact solutions for temperature and velocity fields by Caputo time fractional derivatives in dimensionless form. Mahsud, Shah and Vieru [32] studied unsteady flows of an upper-convected Maxwell fluid which is described by the fractional differential equations with time-fractional Caputo-Fabrizio derivatives. Shah etal [33] studied natural convective flows of Prabhakar-like fractional viscoelastic fluids by introducing the generalized fractional constitutive equations, and used the time-fractional Prabhakar derivative to describle the generalized memory effects. J. Shu et al. [34] studied the asymptotic behavior of the solution of the non-autonomous fractional-order stochastic reaction-diffusion equation with multiplicative noise in R. M. Stynes et al. [35] studied the reaction-diffusion problem with Caputo time derivatives, giving a new analysis of the standard finite difference method for this problem. C.B. Huang et al. [36] presented a fully discrete numerical method for computing an approximate solution of fractional reaction-diffusion initial-boundary value problems based on L1 discretization in time and direct discontinuous Galerkin (DDG) finite element in space. V.K. Baranwal et al. [37] proposed a new analytical algorithm for solving a system of highly nonlinear time-fractional order reaction-diffusion equations, a fusion of the variational iterative method and the Adomian decomposition method. S. Ali et al. [38] obtained an approximate solution of the fractional order Cauchy reaction diffusion equation using the optimal homotopy asymptotic method. New numerical schemes for solving nonlinear fractional convection-diffusion equations of order β∈[1,2] were developed by H. Safdari et al. [39]. They proposed locally discontinuous Galerkin methods by adopting linear, quadratic and cubic B-spline basis functions.
The Discontinuous Galerkin (DG) method is between a finite element and a finite volume method, and uses a discontinuous solution space and has high accuracy for any order of accuracy. The main idea of the LDG method is to transform the original higher-order partial differential equations into several equivalent systems of first-order equations by introducing auxiliary variables and then discretize the obtained systems of first-order equations using the DG method [40]. In this paper, we will consider the LDG method based on the generalized numerical flux to solve the time-fractional reaction-diffusion equation with Caputo fractional order derivatives.
In Section 2, some notations and projections are given. In Section 3, we will propose a fully discrete LDG method for the equation (1.1), and prove that the scheme is unconditionally stable and convergent with O(hk+1+(Δt)2−˙α). The correctness of the theoretical analysis is shown in Section 4 with numerical examples. Finally, the conclusion is given in Section 5.
2.
Notations and auxiliary results
Let a=x12<x32<⋯<xN+12=b be partition of Ω=[a,b], denote Ij=[xj−12,xj+12], for j=1,⋯N, and hj=xj+12−xj−12,1≤j≤N,h=max1≤j≤Nhj.
We denote u+j+12=limt→0+u(xj+12+t) and u−j+12=limt→0+u(xj+12−t).
The piecewise-polynomial space Vkh is defined as
Vkh={ϑ:ϑ∈Pk(Ij),x∈Ij,j=1,2,⋯N},
where k is the order of piecewise polynomial.
For any periodic function ϖ, the following projection is used to prove the error estimate. Let ωe=Pδω−ω, that is P,
∫Ijωeϑdx=0,∀ϑ∈Pk−1(Ij),(ωe)(δ)j+12=0.
(2.1)
Lemma 2.1.Let δ≠12. If ω∈Hs+1[a,b], there holds
‖ωe‖+h12‖ωe‖L2(Γh)≤Chmin(k+1,s+1)‖ω‖s+1,
(2.2)
where the bounding constant C>0 is independent of h and ω. Here Γh denotes the set of boundary points of all elements Ij, and
Let the scalar inner product on L2(E) be denoted by (⋅,⋅)E, and the associated norm by ‖⋅‖E. If E=Ω, we drop E. In the paper, C is a positive number that may have different values in different places.
3.
The scheme
We first construct the fully discrete LDG method for the equation (1.1).
We can rewrite the equation (1.1) into the following form:
p=ux,C0Dα(t)tu+σu−px=F(x,t).
(3.1)
Let tn=nΔt=nMT, Δt=tn−tn−1, we approximate the Caputo fractional derivative C0Dα(tn)tu(x,tn):
Selecting the appropriate numerical flux will play a key role in theoretical analysis for the LDG scheme. From the practical aspect, the generalized alternating numerical fluxes have more application than the traditional numerical fluxes [42]. We consider the following generalized alternating numerical fluxes
here we consider the case δ≠12. For δ=12, the property about unique existence and approximation of the generalized Gauss-Radau projection will become complicated [43].
Next, we give the stability analysis of the scheme (3.3).
3.1. Stability analysis
Without loss of generality, we take F=0 in the theoretical analysis. The following stability result for the scheme (3.3) can be obtained.
Theorem 3.1.For periodic or compactly supported boundary conditions, the fully-discrete LDG scheme (3.3) is unconditionally stable, and the numerical solution unh satisfies
‖unh‖≤‖u0h‖,n=1,2,⋯,M.
(3.6)
Proof. Taking the test functions v=unh,w=pnh in the scheme (3.3), and with the fluxes choice (3.4), we obtain
Theorem 3.2.Let u(x,tn) be the exact solution of the problem (1.1), which is sufficiently smooth with bounded derivatives. Let unh be the numerical solution of the fully discrete LDG scheme (3.3) with flux (3.4), and then the following error estimates holds
is chosen such that the exact solution of the equation is u(x,t)=t2cos(2πx).
Choosing a fixed small time step Δt=11000 to avoid contamination of the temporal error, and dividing the space into N elements to form the uniform mesh. The errors in L2-norm and L∞-norm for different values α(t) and δs are demonstrated in Tables 1 and 2 where N=5,10,20,40. A uniform (k+1)-th order of accuracy for piecewise Pk polynomials can be seen.
Table 1.
Accuracy test with generalized numerical fluxes on uniform meshes, δ=0.3,σ=1,M=103,T=1.
Finally, the temporal convergence rate of the scheme (3.3) for α(t)=1+t2 by piecewise P1 polynomials are provided. Taking the spatial mesh size h=1300, and the temporal meshes Δt=15,110,120,140, respectively. One can find that the temporal convergence rate is first order in Figure 1, which is also consistent with the theoretical results.
Figure 1.L2 errors and L∞ errors versus Δt, order for α(t)=1+t2, k=1.
In this paper, a numerical method is investigated to solve the variable-order (VO) fractional reaction-diffusion equation. We obtained the scheme by using the finite difference method in time and the LDG method in space. Based on the generalized alternating numerical fluxes, we prove that the method is unconditionally stable and convergent with O(hk+1+(Δt)2−˙α). Numerical examples demonstrate the accuracy of our theoretical proofs. In the future, we will use this method to solve models for different kinds of partial differential equations.
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgment
This work is supported by the Scientific and Technological Research Projects in Henan Province (212102210612).
References
[1]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Netherlands, 2006.
[2]
C. P. Li, F. H. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, Boca Raton, 2015.
[3]
I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
[4]
G. Andrew, A Method of Analyzing Experimental Results Obtained from Elasto-Viscous Bodies, J. Appl. Phys., 7 (1936), 311–317. https://doi.org/10.1063/1.1745400 doi: 10.1063/1.1745400
[5]
M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl Geophys, 91 (1971), 134–147. https://doi.org/10.1007/BF00879562 doi: 10.1007/BF00879562
[6]
J. H. He, Some applications of nonlinear fractional differential equations and their applications, Bull. Sci. Technol., 15 (1999), 86–90.
[7]
Z. Jiao, Y. Chen, I. Podlubny, Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives, Springer, 2012. https://doi.org/10.1007/978-1-4471-2852-6
[8]
X. Yang, L. Wu, H. Zhang, A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity, Appl. Math. Comput., 457 (2023), 128192. https://doi.org/10.1016/j.amc.2023.128192 doi: 10.1016/j.amc.2023.128192
[9]
X. M. Gu, H. W. Sun, Y. L. Zhao, X. C. Zheng, An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order, Appl. Math. Lett., 120 (2021), 107270. https://doi.org/10.1016/j.aml.2021.107270 doi: 10.1016/j.aml.2021.107270
[10]
S. Chen, J. Shen, L. L. Wang, Generalized Jacobi functions and their applications tofractional differential equations, Math. Comp., 85 (2016), 1603–1638. https://doi.org/10.1090/mcom3035 doi: 10.1090/mcom3035
[11]
S. Guo, L. Mei, Z. Zhang, Y. Jiang, Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction-diffusion equation, Appl. Math. Lett., 85 (2018, ) 157–163. https://doi.org/10.1016/j.aml.2018.06.005
[12]
C. P. Li, F. H. Zeng, F. Liu, Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal., 15 (2012), 383–406. https://doi.org/10.2478/s13540-012-0028-x doi: 10.2478/s13540-012-0028-x
[13]
X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108–2131. https://doi.org/10.1137/080718942 doi: 10.1137/080718942
[14]
Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
[15]
F. Y. Song, C. J. Xu, Spectral direction splitting methods for two-dimensional space fractional diffusion equations, J. Comput. Phys., 299 (2015), 196–214. https://doi.org/10.1016/j.jcp.2015.07.011 doi: 10.1016/j.jcp.2015.07.011
[16]
J. Guo, C. Li, H. Ding, Finite difference methods for time sub-diffusion equation with space fourth order, Commun. Appl. Math. Comput., 28 (2014), 96–108.
[17]
J. L. Gracia, M. Stynes, Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math., 273 (2015), 103–115. https://doi.org/10.1016/j.cam.2014.05.025 doi: 10.1016/j.cam.2014.05.025
[18]
M. Li, X. M. Gu, C. Huang, M. Fei, G. Zhang, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Compu. Phys., 358 (2018), 256–282. https://doi.org/10.1016/j.jcp.2017.12.044 doi: 10.1016/j.jcp.2017.12.044
[19]
E. Sousa, C. Li, A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, Appl. Numer. Math., 90 (2015), 22–37. https://doi.org/10.1016/j.apnum.2014.11.007 doi: 10.1016/j.apnum.2014.11.007
[20]
W. Bu, A. Xiao, W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 72 (2017), 422–441. https://doi.org/10.1007/s10915-017-0360-8 doi: 10.1007/s10915-017-0360-8
[21]
V. J. Ervin, J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Eq., 22 (2006), 558–576. https://doi.org/10.1002/num.20112 doi: 10.1002/num.20112
[22]
L. Feng, P. Zhuang, F. Liu, I. Turner, Y. Gu, Finite element method for space-time fractional diffusion equation, Numer. Algor., 72 (2016), 749–767. https://doi.org/10.1007/s11075-015-0065-8 doi: 10.1007/s11075-015-0065-8
[23]
Y. N. He, W. W. Sun, Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations, SIAM J. Numer. Anal., 45 (2007), 837–869. https://doi.org/10.1137/050639910 doi: 10.1137/050639910
[24]
Y. N. He, J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1351–1359. https://doi.org/10.1016/j.cma.2008.12.001 doi: 10.1016/j.cma.2008.12.001
[25]
H. Wang, D. Yang, S. Zhu, Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations, SIAM J. Numer. Anal., 52 (2014), 1292–1310. https://doi.org/10.1137/130932776 doi: 10.1137/130932776
[26]
B. Cockburn, C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440–2463. https://doi.org/10.1137/S0036142997316712 doi: 10.1137/S0036142997316712
[27]
L. Guo, Z. Wang, S. Vong, Fully discrete local discontinuous Galerkin methods for some time-fractional fourth-order problems, Int. J. Comput. Math., 93 (2016), 1665–1682. https://doi.org/10.1080/00207160.2015.1070840 doi: 10.1080/00207160.2015.1070840
[28]
Y. Liu, M. Zhang, H. Li, J.C. Li, High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional sub-diffusion equation, Comput. Math. Appl., 73 (2017), 1298–1314. https://doi.org/10.1016/j.camwa.2016.08.015 doi: 10.1016/j.camwa.2016.08.015
[29]
L. Wei, Y. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Model., 38 (2014), 1511–1522. https://doi.org/10.1016/j.apm.2013.07.040 doi: 10.1016/j.apm.2013.07.040
[30]
L. Wei, X. Wei, B. Tang, Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation, Electronic. Res. Arch., 30 (2022), 1263–1281. https://doi.org/10.3934/era.2022066 doi: 10.3934/era.2022066
[31]
M. A. Imran, N. A. Shah, I. Khan, M. Aleem, Applications of non-integer Caputo time fractional derivatives to natural convection flow subject to arbitrary velocity and Newtonian heating, Neural Comput & Applic., 30 (2018), 1589–1599. https://doi.org/10.1007/s00521-016-2741-6 doi: 10.1007/s00521-016-2741-6
[32]
Y. Mahsud, N. A. Shah, D. Vieru, Influence of time-fractional derivatives on the boundary layer flow of Maxwell fluids, Chinese J. Phys., 55 (2017), 1340–1351. https://doi.org/10.1016/j.cjph.2017.07.006 doi: 10.1016/j.cjph.2017.07.006
[33]
N. A. Shah, C. Fetecau, D. Vieru, Natural convection flows of Prabhakar-like fractional Maxwell fluids with generalized thermal transport, J. Therm. Anal. Calorim., 143 (2021), 2245–2258. https://doi.org/10.1007/s10973-020-09835-0 doi: 10.1007/s10973-020-09835-0
[34]
J. Shu, Q. Q. Bai, X. Huang, J. Zhang, Finite fractal dimension of random attractors for non-autonomous fractional stochastic reaction-diffusion equations in R, Appl. Anal., (2020), 1–22.
[35]
M. Stynes, E. O'Riordan, J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
[36]
C. Huang, M. Stynes, A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition, Appl. Numer. Math., (2018). https://doi.org/10.1016/j.apnum.2018.08.006
[37]
V. K. Baranwal, R. K. Pandey, M. P. Tripathi, O. P. Singh, An analytic algorithm for time fractional nonlinear reaction-diffusion equation based on a new iterative method, Commun Nonlinear Sci Numer Simulat, 17 (2012), 3906–3921. https://doi.org/10.1016/j.cnsns.2012.02.015 doi: 10.1016/j.cnsns.2012.02.015
[38]
S. Ali, S. Bushnaq, K. Shah, M. Arif, Numerical treatment of fractional order Cauchy reaction diffusion equations, Chaos, Solitons and Fractals, 103 (2017), 578–587. https://doi.org/10.1016/j.chaos.2017.07.016 doi: 10.1016/j.chaos.2017.07.016
[39]
H. Safdari, M. Rajabzadeh, M. Khalighi, LDG approximation of a nonlinear fractional convection-diffusion equation using B-spline basis functions, Appl. Numer. Math., 171 (2022), 45–57. https://doi.org/10.1016/j.apnum.2021.08.014 doi: 10.1016/j.apnum.2021.08.014
[40]
Y. Xu, C. W. Shu, Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation, Commun. Comput. Phys., 10 (2011), 474–508. https://doi.org/10.4208/cicp.300410.300710a doi: 10.4208/cicp.300410.300710a
[41]
C. B. Huang, M. Stynes, Optimal spatial H1-norm analysis of a finite element method for a time-fractional diffusion equation, J. Comput. Appl. Math., 367 (2020), 112435. https://doi.org/10.1016/j.cam.2019.112435 doi: 10.1016/j.cam.2019.112435
[42]
Y. Cheng, Q. Zhang, H. Wang, Local analysis of the local discontinuous Galerkin method with the generalized alternating numerical flux for two-dimensional singularly perturbed problem, Int. J. Numer. Anal. Modeling, 15 (2018), 785–810.
[43]
Y. Cheng, X. Meng, Q. Zhang, Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp., 86 (2017), 1233–1267. https://doi.org/10.1090/mcom/3141 doi: 10.1090/mcom/3141
Lijie Liu, Xiaojing Wei, Leilei Wei. A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative[J]. Electronic Research Archive, 2023, 31(9): 5701-5715. doi: 10.3934/era.2023289
Lijie Liu, Xiaojing Wei, Leilei Wei. A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative[J]. Electronic Research Archive, 2023, 31(9): 5701-5715. doi: 10.3934/era.2023289