Adaptive Improvements to the LGMRES(m, l) Method for Efficient Resolution of Sparse Linear Systems in Computational Fluid Dynamics

Resumo

Accurate and efficient resolution of sparse linear systems arising from computational fluid dynamics simulations is a computationally intensive task. Traditional iterative methods such as GMRES(m) often suffer from stagnation when applied to large, non-symmetric matrices derived from discretized fluid dynamics models. To address this limitation, a modified version of GMRES(m), known as LGMRES(m,l), was developed to incorporate additional error vectors in the search space. However, previous studies have mainly focused on increasing the subspace dimension m while keeping the number of error vectors l fixed. From a computational cost perspective, this approach may be inefficient, as larger subspaces demand increased memory and processing time, without necessarily enhancing convergence rates.To address this gap, the present work proposes an adaptive enhancement to the LGMRES(m,l) method, incorporating a control-based strategy for dynamic adjustment of both parameters. A proportional control mechanism adjusts m based on the observed rate of convergence, allowing increased subspace dimensions in phases of slow convergence and reduction during fast convergence. Currently, the parameter l is dynamically updated by evaluating the magnitude of error vectors, retaining only those that contribute significantly to residual reduction. This dual adaptation framework aims to mitigate stagnation while optimizing computational resources.The proposed adaptive LGMRES is tested on matrices derived from computational fluid dynamics applications, including groundwater flow models, contaminant transport simulations, and atmospheric modeling for validation. Benchmark matrices from the SuiteSparse Matrix Collection specific to these applications will be used to assess the robustness and efficiency of the adaptive LGMRES method compared to its static counterpart and other iterative solvers. Numerical results obtained using benchmark matrices derived from environmental fluid dynamics applications confirm the effectiveness of the proposed adaptive LGMRES. Significant reductions in the number of iterations and the overall execution time were observed, highlighting the potential of adaptive parameter tuning to improve the robustness and scalability of LGMRES in contexts of computational fluid dynamics.