# A parareal approach of semi‐linear parabolic equations based on general waveform relaxation

@article{Li2019APA, title={A parareal approach of semi‐linear parabolic equations based on general waveform relaxation}, author={Jun Li and Yaolin Jiang and Zhen Miao}, journal={Numerical Methods for Partial Differential Equations}, year={2019}, volume={35}, pages={2017 - 2043} }

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…

## 7 Citations

### Some convergence results of waveform relaxation for a class of second-order quasilinear parabolic equations

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Some superlinear and linear convergence results under several convergence conditions for these quasilinear equations are given, and numerical experiments illustrated these superlinear or linear convergence rates, which claim that waveform relaxation method is available as an iterative framework for nonlinear problems.

### A hybrid algorithm based on parareal and Schwarz waveform relaxation

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A hybrid algorithm based on parareal and Schwarz waveform relaxation (SWR) for solving time dependent partial differential equations and a convergence analysis for the hybrid algorithm for a 1D model problem, the reaction-diffusion equation is given.

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This work extends the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations to solve partial differential examples with higher dimensional, fractional, and nonlinear terms.

### Fast simulation of dynamic heat transfer through building envelopes via parareal algorithms

- EngineeringScience and Technology for the Built Environment
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In this article, two efficient time-parallel algorithms, the classical parareal algorithm and the parareal waveform relaxation (PWR) algorithm, were constructed and investigated for dynamic heat…

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