Structure preserving spectral methods and exponential integrators for the numerical solution of stiff semi-linear partial differential equations
Bibliographic record
Abstract
In this thesis, we present numerical solution of semilinear partial differential equations (PDEs) where the linear differential operator is a self-adjoint. A recent spectral method for self-adjoint operators, based on basis recombination, leads to symmetric definite matrices, which have real spectrum. This allows for developing stable time-stepping algorithm to solve the resulting the ordinary differential equations. The linear part of the discretized problem is usually stiff, which constraints the step size for explicit numerical schemes. We therefore use exponential integrators, a well-known time-stepping methods for solving stiff differential equations. We describe three methods namely, eigen-decomposition, contour integral and Carath\'{e}odory-Fej\'{e}r approximation, for computing the matrix ($\varphi$) functions of the exponential integrators. We perform numerical experiments with some PDEs with different boundary conditions, including time-dependent boundary condition and the numerical results confirm the accuracy of the methods in both space and time.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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".