W-methods to stabilize standard explicit Runge-Kutta methods in the time integration of advection-diffusion-reaction PDEs

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• 作者：S. Gonzá ; lez-Pinto ; ¹ ; ^{spinto@ull.es} ; D. Herná ; ndez-Abreu¹ ; ^{dhabreu@ull.edu.es} ; S. Pé ; rez-Rodrí ; guez¹ ; ^{sperezr@ull.es}
• 关键词：65M12 ; 65M15 ; 65M20
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：15 May 2017
• 年：2017
• 卷：316
• 期：Complete
• 页码：143-160
• 全文大小：635 K
• 卷排序：316

文摘

A technique to stabilize standard explicit Runge–Kutta methods by associating them with W-methods is proposed. The main point to get the associated family of W-methods for a given explicit Runge–Kutta method is to require commutativity for the coefficient matrices of the W-method in order to reduce the large number of order conditions that must be satisfied to get a pre-fixed order.Based on this idea, for any given explicit four-stage Runge–Kutta method of order four, two uniparametric families of third order W-methods are obtained. The free parameter can be used to increase the stability regions of the W-methods in case of d≥1d≥1 splittings in the derivative function when a von Neumann stability analysis is carried out. Additionally, it is possible to find L-stable ROW-methods (W-methods with exact Jacobian) for some specific values of the free parameter.The new family of W-methods is also equipped with the splitting provided by the Approximate Matrix Factorization (AMF), which converts a W-method into some kind of ADI-method (Alternating Direction Implicit method). The AMF-W-methods so obtained are mainly used to solve large time-dependent PDE systems (in 2D or 3D spatial variables) discretized in space by using finite differences or finite volumes. Some stability properties of the family of AMF-W-methods are also supplied for the case of dd-splittings (d≥1d≥1).Numerical experiments in connection with the proposed AMF-W-methods on a few interesting stiff problems coming from PDE discretizations illustrate the stabilization approach in comparison with some relevant methods in the literature.