DOI number:10.1007/s10444-026-10286-8
Journal:Advances in Computational Mathematics
Abstract:In this paper, we explore a direct parallel-in-time method based on the diagonalization of the time-stepping matrix to solve the Kawarada equation, which arises from the quenching combustion process. Unlike the traditional time-marching finite difference method, where a nonlinear system may need to be solved at each time step once implicit schemes are employed, here, we form a large sparse linear system involving all unknown variables on the time-space domain, so that numerical solution can be handled all at once. Thanks to the non-uniform step sizes, the diagonalization technique can be introduced into a parallel-in-time process. Therefore, the dynamics near quenching time and quenching point can be described more accurately.We use a time window technique to release the deficiency of diagonalization, leading the accuracy and efficiency of the algorithm to be well-balanced. In numerical analysis, we propose a practical PAMS-framework to verify the effectiveness of the aforementioned method from four aspects including Positivity, Asymptoticity, Monotonicity, and Stability. Finally, several numerical experiments are conducted on 1D and 2D Kawarada equations, which demonstrate that the studied parallel-in-time method is reliable and highly efficient for quenching-type reaction diffusion equations. Meanwhile, computational time is reduced satisfactorily compared to the time-marching method.
First Author:Yufeng Xu*, Ying Zhu*, Desong Kong**, Zhoushun Zheng*
Indexed by:Journal paper
Discipline:Natural Science
First-Level Discipline:Mathematics
Document Type:J
Volume:52
Issue:15
Page Number:1--28
Translation or Not:no
Included Journals:SCI
Links to published journals:https://doi.org/10.1007/s10444-026-10286-8