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A parareal approach of semi‐linear parabolic equations based on general waveform relaxation
Author(s) -
Li Jun,
Jiang YaoLin,
Miao Zhen
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22390
Subject(s) - mathematics , boundary value problem , ordinary differential equation , relaxation (psychology) , partial differential equation , convergence (economics) , discretization , mathematical analysis , parabolic partial differential equation , ode , initial value problem , waveform , differential equation , computer science , psychology , social psychology , economic growth , economics , radar , telecommunications
We present a parareal approach of semi‐linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial‐boundary value problem and two algorithms for time‐periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial‐boundary value problem is superlinearly convergent while both algorithms for the time‐periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.