Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge–Kutta Methods
Abstract
1. Introduction
2. Preliminaries: Main Tools and Requirements
2.1. The MGRIT Algorithm
Algorithm 1 Two-grid method. |
|
2.2. Tolerance Proportionality and Computational Stability
2.3. Classical Recursive Controllers
3. Results
3.1. ODE Test Problems
3.2. Results for Refinement Factors r, , and
3.2.1. The Final Algorithm with
- (A1)
- Set an initial temporal grid and an initial guess of the solution (see Figure 3).
- (A2)
- Apply a given Runge–Kutta method in parallel from these initial values, and use multigrid cycles to improve the current approximation of the solution (see Figure 2).
- (A3)
- Determine the refinement factor for each subinterval on the finest grid based on (9).
- (A4)
- IF no refinement occurred and the accuracy criterion is satisfied then STOP.
- (A5)
- ELSE go to Step 2.
3.2.2. An Attempt with Exponential-Forgetting Filters
3.2.3. Tolerance Proportionality for the Algorithm (A1)–(A5)
3.3. PDE Test Problems and Related Results
4. Discussion
4.1. Advantages
4.2. Limitations
4.3. Future Research
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
MGRIT | Multigrid reduction in time |
ODE | Ordinary differential equation |
PDE | Partial differential equation |
Tolerance parameter |
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Fekete, I.; Izsák, F.; Kupás, V.P.; Söderlind, G. Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge–Kutta Methods. Algorithms 2025, 18, 484. https://doi.org/10.3390/a18080484
Fekete I, Izsák F, Kupás VP, Söderlind G. Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge–Kutta Methods. Algorithms. 2025; 18(8):484. https://doi.org/10.3390/a18080484
Chicago/Turabian StyleFekete, Imre, Ferenc Izsák, Vendel P. Kupás, and Gustaf Söderlind. 2025. "Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge–Kutta Methods" Algorithms 18, no. 8: 484. https://doi.org/10.3390/a18080484
APA StyleFekete, I., Izsák, F., Kupás, V. P., & Söderlind, G. (2025). Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge–Kutta Methods. Algorithms, 18(8), 484. https://doi.org/10.3390/a18080484