A new framework for solving partial differential equations using semi-analytical explicit RK(N)-type integrators

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• 作者：Xinyuan Wu ; ^{xywu@nju.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Changying Liu ^{chyliu88@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Lijie Mei ^{bxhanm@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65M99 ; 35C15 ; 35K05 ; 35L03 ; 35L05 ; 65L06
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：301
• 期：Complete
• 页码：74-90
• 全文大小：2566 K

文摘

This paper develops a new explicit semi-analytical approach to solving PDEs based on the variation-of-constants formula, or the equivalent integration equation. These new schemes avoid the discretization of spatial derivatives. Therefore, the accuracy of the semi-analytical explicit scheme depends entirely on time integration. We first establish a semi-analytical formula for wave partial differential equations and show the consistency of the semi-analytical formula with different boundary conditions. We then present semi-analytical explicit RKN-type integrators for solving wave PDEs. This methodology is also extended to dealing with first-order hyperbolic PDEs and parabolic PDEs, and for both of them the corresponding semi-analytical explicit RK-type integrators are derived as well. Numerical simulations are implemented and the results show the effectiveness of the new semi-analytical explicit schemes. Meanwhile, an operator-variation-of-constants formula with applications is introduced for high-dimensional nonlinear wave equations before presenting the idea of semi-analytical explicit RKN integrators.