Improving Initial Guess for the Iterative Solution of Linear Equation Systems in Incompressible Flow

Ye, Shuai; Lin, Yufei; Xu, Liyang; Wu, Jin−Jei

doi:10.3390/math8010119

Cited by 3 publications

(4 citation statements)

References 35 publications

(37 reference statements)

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“…1a is exact, and Fig. 1b is the result from GMRES with a good initial guess which is the solution from the previous computation stage [14]. The early stage of Fig.…”

Section: Description Of Cold Start and Its Solutionmentioning

confidence: 99%

Parallel Iterative FEM Solver with Initial Guess for Frequency Domain Electromagnetic Analysis

Lee¹,

Park²,

Park³

et al. 2023

Intelligent Automation &Amp; Soft Computing

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The finite element method is a key player in computational electromagnetics for designing RF (Radio Frequency) components such as waveguides. The frequency-domain analysis is fundamental to identify the characteristics of the components. For the conventional frequency-domain electromagnetic analysis using FEM (Finite Element Method), the system matrix is complex-numbered as well as indefinite. The iterative solvers can be faster than the direct solver when the solver convergence is guaranteed and done in a few steps. However, such complex-numbered and indefinite systems are hard to exploit the merit of the iterative solver. It is also hard to benefit from matrix factorization techniques due to varying system matrix parts according to frequency. Overall, it is hard to adopt conventional iterative solvers even though the system matrix is sparse. A new parallel iterative FEM solver for frequency domain analysis is implemented for inhomogeneous waveguide structures in this paper. In this implementation, the previous solution of the iterative solver of Matlab (Matrix Laboratory) employing the preconditioner is used for the initial guess for the next step's solution process. The overlapped parallel stage using Matlab's Parallel Computing Toolbox is also proposed to alleviate the cold starting, which ruins the convergence of early steps in each parallel stage. Numerical experiments based on waveguide structures have demonstrated the accuracy and efficiency of the proposed scheme.

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“…1a is exact, and Fig. 1b is the result from GMRES with a good initial guess which is the solution from the previous computation stage [14]. The early stage of Fig.…”

Section: Description Of Cold Start and Its Solutionmentioning

confidence: 99%

Parallel Iterative FEM Solver with Initial Guess for Frequency Domain Electromagnetic Analysis

Lee¹,

Park²,

Park³

et al. 2023

Intelligent Automation &Amp; Soft Computing

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show abstract

“…It can be seen from last section that the stability condition

{\overset{true‾}{C}}_{CFL} \leq {\overset{true‾}{C}}_{CFL, cr} = 0.01

should be strictly satisfied when using PISO algorithm, otherwise, poor solutions together with the violation of conservation can be occurred. The PIMPLE algorithm offers more robustness and efficiency as it provides the outer iteration, which is controlled by specifying the

NOC

, to improve the stability 76 . Cases 1 and 6 in Table 2 are selected to further investigate the effect of

NOC

on numerical stability.…”

Section: Stability Analysismentioning

confidence: 99%

“…The PIMPLE algorithm offers more robustness and efficiency as it provides the outer iteration, which is controlled by specifying the NOC, to improve the stability. 76 Cases 1 and 6 in Table 2 are selected to further investigate the effect of NOC on numerical stability. The configuration of these cases regarding internal grids, BSDs, and growth rates keeps unchanged, while the rest of the setup is updated in Table 3.…”

Section: Stability Condition For Pimple Algorithmmentioning

confidence: 99%

Stability analysis of discrete population balance model for bubble growth and shrinkage

Liao

Lucas

et al. 2021

Numerical Methods in Fluids

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The stability condition for solving the population balance equation (PBE) involving bubble growth and shrinkage within the Eulerian framework is proposed. The particle flux weighted average Courant number, Ctrue‾CFL, which reflects the propagation of particle state of the entire size classes on internal coordinates is first derived. Stability conditions of internal convection for PISO and PIMPLE algorithms are obtained by evaluating a series of tests concerning the constant bubble growth. The proposed stability conditions are then applied to simulate two kinds of laboratory experiments, that is, the bubble growth in stagnant liquid and condensing steam‐water pipe flows. The results show that the stable PBE solutions hold for the cases using either PISO or PIMPLE algorithms combined with the corresponding stability condition. Meanwhile, the conservation of the zeroth and first moments of the number density function can be guaranteed when the stability condition is satisfied. In addition, the effect of heat transfer coefficient correlations is also discussed.

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“…This can be exploited to construct a better initial guess, either explicitly through (polynomial) extrapolation, or implicitly through projection techniques. Examples of the extrapolation approach include polynomial POD extrapolation [14], weighted group extrapolation methods [30] and a stabilized, least-squares polynomial extrapolation method [1], for the case that only the right-hand side evolves in time. For the same setting, projection techniques have been introduced by Fischer [11].…”

Section: Introductionmentioning

confidence: 99%

Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

Guido,

Kressner,

Ricci

2024

J Sci Comput

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We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.

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Improving Initial Guess for the Iterative Solution of Linear Equation Systems in Incompressible Flow

Cited by 3 publications

References 35 publications

Parallel Iterative FEM Solver with Initial Guess for Frequency Domain Electromagnetic Analysis

Parallel Iterative FEM Solver with Initial Guess for Frequency Domain Electromagnetic Analysis

Stability analysis of discrete population balance model for bubble growth and shrinkage

Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

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