A generalization of the Runge-Kutta iteration

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:224

Article

Haelterman, R.¹  Vierendeels, J.²  Van Heule, D.¹ 

[1] Royal Mil Acad, Dept Math MWMW, B-1000 Brussels, Belgium

[2] Univ Ghent, Dept Flow Heat & Combust Mech, B-9000 Ghent, Belgium

关键词: Iterative solution;    Multi-grid;    Multi-stage;   

DOI  :  10.1016/j.cam.2008.04.021

来源: Elsevier

【 摘 要 】

Iterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge-Kutta iteration. We show that a generally used formulation [A. Jameson, Numerical solution of the Euler equations for compressible inviscid fluids, in: F. Angrand, A. Dervieux. J.A. Desideri, R. Glowinski (Eds.), Numerical Methods for the Euler Equations of Fluid Dynamics, SIAM, Philadelphia, 1985, pp. 199-245] does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this, without using an excessive number of coefficients. After having converted the optimal parameters found in previous studies (e.g. [B. Van Leer, C.H. Tai, K.G. Powell, Design of optimally smoothing multi-stage schemes for the Euler equations, AIAA Paper 89-1923, 1989]) we compare them with those that we obtain when we optimize for an integrated 2-grid V-cycle and show that this results in Superior performance using a low number of stages. We also propose a variant of our new formulation that roughly follows the idea of the Martinelli-Jameson scheme [A. Jameson, Analysis and design of numerical schemes for gas dynamics 1, artificial diffusion, upwind biasing, limiter and their effect on multigrid convergence, Int. J. Comput. Fluid Dyn. 4 (1995) 171-218: J.V. Lassaline, Optimal multistage relaxation coefficients for multigrid flow solvers. http://www.ryerson.ca/similar to jvl/papers/cfd2005.pdf] used on the advection-diffusion equation which that can be extended to other types. Gains in the order of 30%-50% have been shown with respect to classical iterative schemes on the advection equation. Better results were also obtained on the advection-diffusion equation than with the Martinelli-Jameson coefficients, but with less than half the number of matrix-vector multiplications. (C) 2008 Elsevier B.V. All rights reserved.

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