A second-order difference scheme for the time fractional substantial diffusion equation

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• 作者：Zhaopeng Hao^a ; ^{230139121@seu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Wanrong Cao^a ; ^{wrcao@seu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Guang Lin^b ; ^{guanglin@purdue.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：26A33 ; 65M06 ; 65M12 ; 65M55 ; 65T50
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：313
• 期：Complete
• 页码：54-69
• 全文大小：508 K

文摘

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Grünwald formula and its asymptotic expansion. Moreover, the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time. With the use of the fourth-order compact scheme in space, we give a fully discrete Grünwald–Letnikov-formula-based compact difference scheme and prove its stability and convergence by the energy method under smooth assumptions. In addition, the problem with nonsmooth solution is also discussed, and an improved algorithm is proposed to deal with the singularity of the fractional substantial derivative. Numerical examples show the reliability and efficiency of the scheme.