Numerical simulations of time-dependent partial differential equations

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• 作者：Francisco de la Hoz ^{francisco.delahoz@ehu.es" class="auth_mail" title="E-mail the corresponding author} ; Fernando Vadillo ; ^{fernando.vadillo@ehu.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65M70 ; 65M20
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：295
• 期：Complete
• 页码：175-184
• 全文大小：1526 K

文摘

When a time-dependent partial differential equation (PDE) is discretized in space with a spectral approximation, the result is a coupled system of ordinary differential equations (ODEs) in time. This is the notion of the method of lines (MOL), and the resulting set of ODEs is stiff; the stiffness may be even exacerbated sometimes. The linear terms are the primarily responsible for the stiffness, with a rapid exponential decay of some modes (as in a dissipative PDE), or a rapid oscillation of some modes (as in a dispersive PDE). Therefore, for a time-dependent PDE which combines low-order nonlinear terms with higher-order linear terms, it is desirable to use a higher-order approximation both in space and in time.

Along our research, we have focused on a particular case of spectral methods, the so-called pseudo-spectral methods, to solve numerically time-dependent PDEs using different techniques: an integrating factor, in de la Hoz and Vadillo (2010); an exponential time differencing method, in de la Hoz and Vadillo (2008); and differentiation matrices in the theoretical frame of matrix differential equations, in de la Hoz and Vadillo (2012, 2013a,b). This paper, which is a unified review of those contributions, aims at providing a better understanding of those methods, by illustrating their variety and, more importantly, their power. Furthermore, we also give emphasis to choosing adequate schemes to advance in time.