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
Bibliographic record
Abstract
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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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".