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A Comparison of Sparse Solvers for Severely Ill-Conditioned Linear Systems in Geophysical Marker-In-Cell Simulations arxiv.org/abs/2409.11515

A Comparison of Sparse Solvers for Severely Ill-Conditioned Linear Systems in Geophysical Marker-In-Cell Simulations

Solving sparse linear systems is a critical challenge in many scientific and engineering fields, particularly when these systems are severely ill-conditioned. This work aims to provide a comprehensive comparison of various solvers designed for such problems, offering valuable insights and guidance for domain scientists and researchers. We develop the tools required to accurately evaluate the performance and correctness of 16 solvers from 11 state-of-the-art numerical libraries, focusing on their effectiveness in handling ill-conditioned matrices. The solvers were tested on linear systems arising from a coupled hydro-mechanical marker-in-cell geophysical simulation. To address the challenge of computing accurate error bounds on the solution, we introduce the Projected Adam method, a novel algorithm to efficiently compute the condition number of a matrix without relying on eigenvalues or singular values. Our benchmark results demonstrate that Intel oneAPI MKL PARDISO, UMFPACK, and MUMPS are the most reliable solvers for the tested scenarios. This work serves as a resource for selecting appropriate solvers, understanding the impact of condition numbers, and improving the robustness of numerical solutions in practical applications.

arxiv.org

Error estimation for numerical approximations of ODEs via composition techniques. Part I: One-step methods arxiv.org/abs/2409.10548

Error estimation for numerical approximations of ODEs via composition techniques. Part I: One-step methods

In this study, we introduce a refined method for ascertaining error estimations in numerical simulations of dynamical systems via an innovative application of composition techniques. Our approach involves a dual application of a basic one-step numerical method of order p in this part, and for the class of Backward Difference Formulas schemes in the second part [Deeb A., Dutykh D. and AL Zohbi M. Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods, Submitted, 2024]. This dual application uses complex coefficients, resulting outputs in the complex plane. The methods innovation lies in the demonstration that the real parts of these outputs correspond to approximations of the solutions with an enhanced order of p + 1, while the imaginary parts serve as error estimations of the same order, a novel proof presented herein using Taylor expansion and perturbation technique. The linear stability of the resulted scheme is enhanced compared to the basic one. The performance of the composition in computing the approximation is also compared. Results show that the proposed technique provide higher accuracy with less computational time. This dual composition technique has been rigorously applied to a variety of dynamical problems, showcasing its efficacy in adapting the time step,particularly in situations where numerical schemes do not have theoretical error estimation. Consequently, the technique holds potential for advancing adaptive time-stepping strategies in numerical simulations, an area where accurate local error estimation is crucial yet often challenging to obtain.

arxiv.org
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