scholarly journals Fully Discrete Local Discontinuous Galerkin Approximation for Time-Space Fractional Subdiffusion/Superdiffusion Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Meilan Qiu ◽  
Liquan Mei ◽  
Dewang Li
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Second Order ◽  
Laplacian Operator ◽  
Numerical Schemes ◽  
Central Difference ◽  
Fully Discrete ◽  
Local Discontinuous Galerkin ◽  
Time Space ◽  
Backward Euler

We focus on developing the finite difference (i.e., backward Euler difference or second-order central difference)/local discontinuous Galerkin finite element mixed method to construct and analyze a kind of efficient, accurate, flexible, numerical schemes for approximately solving time-space fractional subdiffusion/superdiffusion equations. Discretizing the time Caputo fractional derivative by using the backward Euler difference for the derivative parameter (0<α<1) or second-order central difference method for (1<α<2), combined with local discontinuous Galerkin method to approximate the spatial derivative which is defined by a fractional Laplacian operator, two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusion/superdiffusion equations are proposed, respectively. Through the mathematical induction method, we show the concrete analysis for the stability and the convergence under theL2norm of the LDG schemes. Several numerical experiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes. The numerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerful for solving fractional partial differential equations.

scholarly journals Local Discontinuous Galerkin Method for Nonlinear Time-Space Fractional Subdiffusion/Superdiffusion Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Meilan Qiu ◽  
Dewang Li ◽  
Yanyun Wu
Keyword(s):  
Convergence Rate ◽  
Fractional Derivative ◽  
Discontinuous Galerkin ◽  
Numerical Schemes ◽  
Step Size ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Time Space ◽  
Backward Euler ◽  
Heterogeneous Rocks

Fractional partial differential equations with time-space fractional derivatives describe some important physical phenomena. For example, the subdiffusion equation (time order 0<α<1) is more suitable to describe the phenomena of charge carrier transport in amorphous semiconductors, nuclear magnetic resonance (NMR) diffusometry in percolative, Rouse, or reptation dynamics in polymeric systems, the diffusion of a scalar tracer in an array of convection rolls, or the dynamics of a bead in a polymeric network, and so on. However, the superdiffusion case (1<α<2) is more accurate to depict the special domains of rotating flows, collective slip diffusion on solid surfaces, layered velocity fields, Richardson turbulent diffusion, bulk-surface exchange controlled dynamics in porous glasses, the transport in micelle systems and heterogeneous rocks, quantum optics, single molecule spectroscopy, the transport in turbulent plasma, bacterial motion, and even for the flight of an albatross (for more physical applications of fractional sub-super diffusion equations, one can see Metzler and Klafter in 2000). In this work, we establish two fully discrete numerical schemes for solving a class of nonlinear time-space fractional subdiffusion/superdiffusion equations by using backward Euler difference 1<α<2 or second-order central difference 1<α<2/local discontinuous Galerkin finite element mixed method. By introducing the mathematical induction method, we show the concrete analysis for the stability and the convergence rate under the L2 norm of the two LDG schemes. In the end, we adopt several numerical experiments to validate the proposed model and demonstrate the features of the two numerical schemes, such as the optimal convergence rate in space direction is close to Ohk+1. The convergence rate in time direction can arrive at Oτ2−α when the fractional derivative is 0<α<1. If the fractional derivative parameter is 1<α<2 and we choose the relationship as h=C′τ (h denotes the space step size, C′ is a constant, and τ is the time step size), then the time convergence rate can reach to Oτ3−α. The experiment results illustrate that the proposed method is effective in solving nonlinear time-space fractional subdiffusion/superdiffusion equations.

A local discontinuous Galerkin approximation for systems with p-structure

2013 ◽  
Vol 34 (4) ◽  
pp. 1447-1488 ◽  
Author(s):  
L. Diening ◽  
D. Kroner ◽  
M. R  i ka ◽  
I. Toulopoulos
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Local Discontinuous Galerkin
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