Abstract
Multiparameter extensions (MP) of (linear and nonlinear) descent methods have been proposed for the solution of finite dimensional time independent problems; these new methods are based on a different treatment of several blocks of components of the solution, basically via the substitution of a scalar relaxation by a (suitable) matricial relaxation. Similarly, the Nonlinear Galerkin Method (NLG), that stems from the dynamical system theory, propose to apply distinct temporal integration schemes to different sets of data scales when solving dissipative PDEs. In this paper, the algebraic similarity of Richardson iteration and Forward-Euler time integration is extended to new grounds through the expansion of the realm of MP methods to the field of the numerical integration of dissipative PDEs. The separation of the structures is realized by the utilization of hierarchical preconditioners in finite differences, which are conjugated to a MP temporal integration steeming from NLG theory. Numerical examples of fluid dynamics problems show the improved temporal stability of these new methods as compared to the classical ones.
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Chehab, J., Costa, B. Multiparameter Schemes for Evolutionary Equations. Numerical Algorithms 34, 245–257 (2003). https://doi.org/10.1023/B:NUMA.0000005401.91113.1f
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DOI: https://doi.org/10.1023/B:NUMA.0000005401.91113.1f