Multi-Level and Time-Parallel Solution of Time-Periodic Problems / submitted by Sergiy Bogdanov, M.Sc. Wuppertal, April 13, 2025
Inhalt
- List of Abbreviations
- 1 Introduction
- 1.1 Evolution of Iterative Methods in Computational Mathematics
- 1.2 The Evolution and Importance of Multi-grid Methods
- 1.3 Research Gap and Objectives
- 1.4 Practical Relevance of the Research
- 1.5 Structure of the Thesis
- 2 Literature Review
- 2.1 Historical Development of Multi-grid Methods
- 2.2 Review of Applications and Case Studies
- 2.3 Emergence of QMGRIT
- 2.4 Critical Analysis of Current Literature
- 2.5 Summary Overview and Research Initiation
- 3 QMGRIT Algorithm Development
- 3.1 Excursus on Eternal Wanderlust and Test Equations
- 3.1.1 Existence and Uniqueness of the Solution
- 3.1.2 Basic Iterative Method
- 3.1.3 Eternal Wanderlust
- 3.1.4 Heat Equation
- 3.1.5 Fourier-Poisson-Kelvin Problem: Diffusion, Convection, and Decay
- 3.1.6 Wave Equation
- 3.2 Excursus on QMGRIT
- 3.2.1 Excursus to the Spectral Analysis of QMGRIT
- 3.2.2 Numerical Analysis Validation
- 3.2.3 Ghosted QMGRIT
- 3.2.4 Alternative Analysis Methodology with SAMA
- 3.2.5 Comparison with periodic Parareal
- 3.3 Intermediate Conclusion
- 4 Numerical Experiments
- 4.1 Coaxial Cable Problem
- 4.1.1 The Partial Differential Equation
- 4.1.2 Derivation of the Analytical Solution
- 4.1.3 Model Parameters
- 4.1.4 Non-linear Model Formulation
- 4.1.5 Numerics of QMGRIT
- 4.2 Electrical Machine
- 4.3 Convection-Diffusion-Decay Equation
- 4.4 Wave Equation
- 4.4.1 Two-variable System — Discretization and Numerical Solution
- 4.4.2 Two-grid Solution
- 4.4.3 Three-grid and Four-grid Solutions
- 4.4.4 Four-grid and Five-grid Solutions — Achieving a `Convenient' Convergence Factor at the Cost of High Computational Overhead
- 4.5 Interplay with QMGRIT on GMRES — Empirical Analysis
- 5 Summary, Conclusions and Future Work
- A LFA vs. SAMA
- B FEM
- List of Algorithms
- List of Figures
- List of Tables
- Bibliography