Arbitrary High-Order Unconditionally Stable Methods for Reaction-Diffusion Equations with inhomogeneous Boundary Condition via Deferred Correction
Bibliographic record
Abstract
Abstract In this paper, we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations. To avoid possible order reduction, the IBVP is first transformed into an IBVP with homogeneous boundary conditions (IBVPHBC) via a lifting of inhomogeneous Dirichlet, Neumann or mixed Dirichlet–Neumann boundary conditions. The IBVPHBC is discretized in time via the deferred correction method for the implicit midpoint rule and leads to a time-stepping scheme of order 2 p + 2 2p+2 of accuracy at the stage p = 0 , 1 , 2 , … p=0,1,2,\ldots of the correction. Each semi-discretized scheme results in a nonlinear elliptic equation for which the existence of a solution is proven using the Schaefer fixed point theorem. The elliptic equation corresponding to the stage 𝑝 of the correction is discretized by the Galerkin finite element method and gives a full discretization of the IBVPHBC. This fully discretized scheme is unconditionally stable with order 2 p + 2 2p+2 of accuracy in time. The order of accuracy in space is equal to the degree of the finite element used when the family of meshes considered is shape-regular, while an increment of one order is proven for a quasi-uniform family of meshes. Numerical tests with a bistable reaction-diffusion equation having a strong stiffness ratio, a Fisher equation, a linear reaction-diffusion equation addressing order reduction and two linear IBVPs in two dimensions are performed and demonstrate the unconditional convergence of the method. The orders 2, 4, 6, 8 and 10 of accuracy in time are achieved. Except for some linear problems, the accuracy of DC methods is better than that of BDF methods of same order.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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 source (direct Gemma or distilled Codex), 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".