Conservation Properties of Finite-Difference Time-Domain Methods for the Maxwell and Schrödinger Equations With Application to the Development of New Schemes with Guaranteed Stability
Notice bibliographique
Résumé
The finite-difference time-domain (FDTD) method is a popular technique for solving the Maxwell equations. The method works by updating unknowns in a leap-frog manner, starting with initial conditions. The update equations are explicit, and hence have relatively low computational requirements. However, the explicit nature comes at the cost of conditional stability, which makes it substantially harder to create new FDTD-based methods, compared to implicit or frequency-domain methods. Methodologies for stability analysis exist, but can lead to lengthy derivations when applied to complex problems. FDTD has also been applied to solving the Schrödinger equation. Research related to this scheme, along with other numerical methods for quantum mechanical problems, is becoming increasingly important. Another type of FDTD methods that were recently inspired by quantum applications are schemes that use potentials as unknowns in order to facilitate modelling of light-matter interaction. Much like in the conventional FDTD, the difficulty of stability analysis poses a significant challenge in advancing these schemes. This thesis draws inspiration from results in control theory in order to facilitate stability analysis of these FDTD methods based on the concepts of energy conservation and dissipation, as well as probability conservation in the case of FDTD for the Schrödinger equation. For the fields-based FDTD, expressions are proposed to quantify the energy stored in a region and the energy entering the region through the boundary. Based on these expressions, we formulate rigorous conditions under which the region is unable to generate spurious energy. These results provide a way to create new schemes by coupling the FDTD equations for the region to other dissipative or energy conserving models. As an example we develop a subgridding scheme by coupling two FDTD grids of different resolution in an energy-conserving manner. The theory is extended to FDTD for the Maxwell equations written in terms of potentials and a subgridding scheme with potentials is developed as a demonstration of application. Furthermore, this thesis extends the approach to propose a similar framework for FDTD for the Schrödinger equation, based on the concept of probability conservation. Conditions for the energy conservation of the scheme are also derived, motivated by the possibility of extending the results to the coupled Maxwell-Schrödinger system of equations in the future.
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Imitation des enseignantsNi prévalence calibrée, ni vérité terrain. Validation humaine à venir. Apprise à partir de 10 348 étiquettes directes de Codex et de 10 348 étiquettes directes de Gemma. Le mode candidate est l'union des têtes enseignantes seuillées; le consensus est leur intersection. Ces sorties portent le statut machine_predicted_unvalidated et ne sont ni des étiquettes humaines ni des étiquettes directes de modèles de pointe.
Scores Codex et Gemma par catégorie
| Catégorie | Codex | Gemma |
|---|---|---|
| Métarecherche | 0,001 | 0,001 |
| Méta-épidémiologie (sens strict) | 0,000 | 0,000 |
| Méta-épidémiologie (sens large) | 0,000 | 0,000 |
| Bibliométrie | 0,000 | 0,001 |
| Études des sciences et des technologies | 0,000 | 0,000 |
| Communication savante | 0,000 | 0,000 |
| Science ouverte | 0,000 | 0,000 |
| Intégrité de la recherche | 0,000 | 0,000 |
| Charge utile insuffisante (le modèle a refusé de juger) | 0,000 | 0,000 |
Scores machine (provisoires)
Les deux têtes enseignantes du modèle étudiant, lues sur ce travail. Un score ordonne la base pour la relecture; il n'affirme jamais une catégorie, et le statut de validation accompagne chaque rangée tel quel.
Scores de référence d'un modèle non mature (critères de maturité non atteints, 7 itérations). Un score ordonne; il n'affirme jamais une catégorie.
score_only:v0-immature-baseline · tel quel depuis la passe de notation : score_only signifie que le nombre peut ordonner les travaux, et qu'aucune étiquette de catégorie n'en découleClassification
machine, non validéePrédiction automatique; un appel candidat d’une seule tête enseignante, pas un consensus.
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