Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations
Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations
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DOI：
10.1137/130918174
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发表时间：
2013-04
期刊：
影响因子：
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通讯作者：
Qinwu Xu;J. Hesthaven
中科院分区：
文献类型：
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作者：
Qinwu Xu;J. Hesthaven
提出了一种求解分数阶对流扩散方程的间断Galerkin方法，通过分数阶拉普拉斯算子定义了$α(1<；α<；2)阶的超扩散算子。将$\α$阶分数次算子表示为一阶导数与$2-\α$阶分数次积分的组合。将分数阶对流扩散问题表示为一组低阶微分/积分方程组，并提出了求解该方程的局部间断Galerkin方法格式。我们证明了分数阶扩散问题的稳定性和最佳收敛阶，并建立了一般分数阶对流扩散问题的最优收敛阶。数值算例验证了分析的正确性。
We propose a discontinuous Galerkin method for fractional convection-diffusion equations with a superdiffusion operator of order $\alpha (1<\alpha<2)$ defined through the fractional Laplacian. The fractional operator of order $\alpha$ is expressed as a composite of first order derivatives and a fractional integral of order $2-\alpha$. The fractional convection-diffusion problem is expressed as a system of low order differential/integral equations, and a local discontinuous Galerkin method scheme is proposed for the equations. We prove stability and optimal order of convergence ${\cal O}(h^{k+1})$ for the fractional diffusion problem, and an order of convergence of ${\cal O}(h^{k+\frac{1}{2}})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.