A Hybridized Discontinuous Galerkin Method for 2D Fractional Convection–Diffusion Equations

A hybridized discontinuous Galerkin method is proposed for solving 2D fractional convection–diffusion equations containing derivatives of fractional order in space on a finite domain. The Riemann–Liouville derivative is used for the spatial derivative. Combining the characteristic method and the hybridized discontinuous Galerkin method, the symmetric variational formulation is constructed. The stability of the presented scheme is proved. Theoretically, the order of $$\mathscr {O}(h^{k+1/2}+\varDelta t)$$O(hk+1/2+Δt) is established for the corresponding models and numerically the better convergence rates are detected by carefully choosing the numerical fluxes. Extensive numerical experiments are performed to illustrate the performance of the proposed schemes. The first numerical example is to display the convergence orders, while the second one justifies the benefits of the schemes. Both are tested with triangular meshes.

[1]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[2]  William McLean,et al.  Fast Summation by Interval Clustering for an Evolution Equation with Memory , 2012, SIAM J. Sci. Comput..

[3]  B. West.  Fractional Calculus in Bioengineering , 2007 .

[4]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[5]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[6]  Jan S. Hesthaven,et al.  Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations , 2013, SIAM J. Numer. Anal..

[7]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[8]  Zhangxin Chen,et al.  Error Analysis for Characteristics-Based Methods for Degenerate Parabolic Problems , 2002, SIAM J. Numer. Anal..

[9]  J. Hesthaven,et al.  Local discontinuous Galerkin methods for fractional diffusion equations , 2013 .

[10]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[11]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[12]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[13]  Vickie E. Lynch,et al.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model , 2001 .

[14]  Zhangxin Chen,et al.  Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems , 2002 .

[15]  CockburnBernardo,et al.  A hybridizable discontinuous Galerkin method for fractional diffusion problems , 2015 .

[16]  Yinnian He,et al.  An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation , 2014, J. Comput. Phys..

[17]  William McLean,et al.  Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation , 2011, Numerical Algorithms.

[18]  William McLean,et al.  Superconvergence of a Discontinuous Galerkin Method for Fractional Diffusion and Wave Equations , 2012, SIAM J. Numer. Anal..

[19]  Kassem Mustapha,et al.  A hybridizable discontinuous Galerkin method for fractional diffusion problems , 2014, Numerische Mathematik.

[20]  W. Deng,et al.  A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation , 2013, 1304.3788.

[21]  J. Kirchner,et al.  Fractal stream chemistry and its implications for contaminant transport in catchments , 2000, Nature.

[22]  William McLean,et al.  Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation , 2009, Numerical Algorithms.

[23]  Zhibo Wang,et al.  A high-order exponential ADI scheme for two dimensional time fractional convection-diffusion equations , 2014, Comput. Math. Appl..

[24]  Mingrong Cui,et al.  A high-order compact exponential scheme for the fractional convection-diffusion equation , 2014, J. Comput. Appl. Math..

[25]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[26]  Huazhong Tang,et al.  High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations , 2012 .

[27]  Weihua Deng,et al.  Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes , 2014, J. Comput. Phys..

[28]  Kassem Mustapha,et al.  Time-stepping discontinuous Galerkin methods for fractional diffusion problems , 2014, Numerische Mathematik.

[29]  Ilaria Perugia,et al.  An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems , 2000, SIAM J. Numer. Anal..

[30]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[31]  Bruce J. West,et al.  Lévy dynamics of enhanced diffusion: Application to turbulence. , 1987, Physical review letters.