The Numerical Solution of Fractional Convection-Diffusion Problems Using a Second-Order Finite Volume Method
Main Article Content
Abstract
The time second-order characteristic finite volume method is proposed for solving the one-dimensional Riemann-Liouville space fractional convection-diffusion equation. To be specific, by employing the Euler-Lagrange integration approach, the fractional convection-diffusion equation is transformed into a parabolic-like equation, simplifying its numerical treatment. To achieve a high level of time accuracy, the second-order Runge-Kutta method is applied to solve the characteristic line equation, while the Crank-Nicholson implicit scheme is employed to handle the discretized equations efficiently. Furthermore, the parabolic-like equation is discretized utilizing piecewise linear finite elements to ensure the spatial accuracy. Then, a detailed analysis of the coefficient matrix for iterative equation reveals favorable numerical properties that enhance the stability and convergence of the proposed scheme. Numerical examples are given to verify the convergence order of our scheme is O(h^(l+alpha)) in space step and O(tao^2) in time step. The results demonstrate the potential of the proposed method as a powerful and effective tool for solving complex fractional convection-diffusion problems in scientific and engineering applications.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
References
A. Bekir, “Exact solutions of some fractional differential equations arising in mathematical biology.” International Journal of Biomathematics, 08 (2018): 155-203.
B. Baeumer, D. Benson, M. Meerschaert, S. Wheatcraft, “Subordinated advection dispersion equation for contaminant transport.” Water Resources Research, 37 (2001): 1543-1550.
J. Kirchner, X. Feng, C. Neal, “Fractal stream chemistry and its implications for contaminant transport in catchments.” Nature, 403 (2000): 524-526.
R. Metzler and T. Nonnenmacher, “Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials.” International Journal of Plasticity, 19 (2003): 941-959.
R. Schumer, D. Benson, M. Meerschaert, B. Baeumer, “Multiscaling fractional advection dispersion equations and their solutions.” Water Resources Research, 39 (2003): 1022-1032.
X. Wang, M. Petru, L. Xia, “Modeling the dynamics behavior of flax fiber reinforced composite after water aging using a modified Huet-Sayegh viscoelastic model with fractional derivatives.” Construction and building Materials, 290 (2021): 12-16.
Y. Zhang and X. J. Yang, “An efficient analytical method for solving local fractional nonlinear PDEs arising in mathematical physics.” Applied Mathematical Modelling, 40 (2015): 1793-1799.
M. Chen and W. Deng, “Fourth order accurate scheme for the space fractional diffusion equations.” SIAM Journal on Numerical Analysis, 52 (2014): 1418-1438.
H. Ding, C. Li, Y. Chen, “High-order algorithms for Riesz derivative and their applications (II).” Journal of Computational Physics, 293 (2015): 218-237.
T. Hang, Z. Zhou, H. Pan, Y. Wang, “The conservative characteristic difference method and analysis for solving two-sided space-fractional advection-diffusion equations.” Numerical Algorithms, 92 (2023), 1723-1755.
S. Vong, P. Lyu, X. Chen, et al., “High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives.” Numerical Algorithms, 72 (2015): 195-210.
W. Tian, H. Zhou, W. Deng, “A class of second order difference approximations for solving space fractional diffusion equations.” Mathematics of Computation, 84 (2015): 1703-1727.
L. Feng, P. Zhuang, F. Liu, I. Turner, Y. Gu, “Finite element method for space-time fractional diffusion equation.” Numerical Algorithms, 72 (2016): 749-767.
D. Nie, J. Sun, W. Deng, “Numerical algorithm for the model describing anomalous diffusion in expanding media.” ESAIM Mathematical Modelling and Numerical Analysis, 54 (2020): 2265-2294.
H. Wang and D. Yang, “Wellposedness of variable-coefficient conservative fractional elliptic differential equations.” SIAM Journal on Numerical Analysis, 51 (2013), 1088-1107.
J. Jia and H. Wang, “A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh.” Journal of Computational Physics, 299 (2015): 842-862.
J. Jia and H. Wang, “A fast finite volume method for conservative space-fractional diffusion equations in convex domains.” Journal of Computational Physics, 310 (2016): 63-84.
F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, “A new fractional finite volume method for solving the fractional diffusion equation.” Applied Mathematical Modelling, 38 (2014): 3871-3878.
Aer. Simmons, Q. Yang, T. Moroney, “A finite volume method for two-sided fractional diffusion equations on non-uniform meshes.” Journal of Computational Physics, 335 (2017): 747-759.
J. Pan, M. Ng, H. Wang, “Fast preconditioned iterative methods for finite volume discretization of steady-state space-fractional diffusion equations.” Numerical Algorithms, 74 (2017): 153-173.
H. Fu, H. Liu, H. Wang, “A finite volume method for two-dimensional Riemann-Liouville space-fractional diffusion equation and its efficient implementation.” Journal of Computational Physics, 388 (2019): 316-334.
X. Zhang, J. Crawford, L. Deeks, M. Stutter, A. Bengough, I. Young, “A mass balance based numerical method for the fractional advection-dispersion equation: theory and application.” Water resources research, 41 (2005): W07029.
J. Douglas Jr. and T. Russell, “Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures.” SIAM Journal on Numerical Analysis, 19 (1982): 871-885.
K. Wang and H. Wang, “A fast characteristic finite difference method for fractional advection-diffusion equations.” Advances in water resources, 34 (2011): 810-816.
H. Rui and M. Tabata, “A mass-conservative characteristic finite element scheme for convection-diffusion problems.” Journal of Scientific Computing, 433 (2010): 416-432.
P. Colella and P. Woodward, “The Piecewise Parabolic Method (PPM) for gas-dynamical simulations.” Journal of computational physics, 54 (1984): 174-201.
K. Fu and D. Liang, “The time second order mass conservative characteristic FDM for advection-diffusion equations in high dimensions.” Journal of Scientific Computing, 73 (2017): 26-49.
K. Fu and D. Liang, “The conservative characteristic FD methods for atmospheric aerosol transport problems.” Journal of Computational Physics, 305 (2016): 494-520.
K. Fu and D. Liang, “A mass-conservative temporal second order and spatial fourth order characteristic finite volume method for atmosphertic pollution advection diffusion problems.” SIAM Journal on Scientific Computing, 41 (2019): 1178-1210.
M. Celia, T. Russell, I. Herrera, R. Ewing, “An Euler-Lagrangian localized adjoint method for the advection-diffusion equations.” Advances in water resources, 13 (1990): 187-206.
H. Dahle, R. Ewing, T. Russell, “Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation.” Computer methods in applied mechanics and engineering, 122 (1995): 223-250.
D. Liang, C. Du, H. Wang, “A fractional step ELLAM approach to high-dimensional convection-diffusion problems with forward particle tracking.” Journal of Computational Physics, 221(2007): 198-225.