Quadratic spline collocation and parareal deferred correction method for parabolic PDEs

2016 ◽  
Author(s):  
Jun Liu ◽  
Yan Wang ◽  
Rongjian Li
Keyword(s):  
Correction Method ◽  
Spline Collocation ◽  
Parabolic Pdes ◽  
Quadratic Spline ◽  
Deferred Correction ◽  
Quadratic Spline Collocation

scholarly journals A Hybrid Algorithm Based on Optimal Quadratic Spline Collocation and Parareal Deferred Correction for Parabolic PDEs

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Jun Liu ◽  
Yan Wang ◽  
Rongjian Li
Keyword(s):  
Hybrid Algorithm ◽  
Computational Cost ◽  
Spline Collocation ◽  
Parabolic Pdes ◽  
Quadratic Spline ◽  
Deferred Correction ◽  
Correction Technique ◽  
The Stability ◽  
The Time Domain ◽  
Quadratic Spline Collocation

Parareal is a kind of time parallel numerical methods for time-dependent systems. In this paper, we consider a general linear parabolic PDE, use optimal quadratic spline collocation (QSC) method for the space discretization, and proceed with the parareal technique on the time domain. Meanwhile, deferred correction technique is also used to improve the accuracy during the iterations. In fact, the optimal QSC method is a correction of general QSC method. Along the temporal direction we embed the iterations of deferred correction into parareal to construct a hybrid method, parareal deferred correction (PDC) method. The error estimation is presented and the stability is analyzed. To save computational cost, we find out a simple way to balance the two kinds of iterations as much as possible. We also argue that the hybrid algorithm has better system efficiency and costs less running time. Numerical experiments by multicore computers are attached to exhibit the effectiveness of the hybrid algorithm.

Optimal superconvergent one step quadratic spline collocation methods

2008 ◽  
Vol 48 (3) ◽  
pp. 449-472 ◽  
Author(s):  
B. Bialecki ◽  
G. Fairweather ◽  
A. Karageorghis ◽  
Q.N. Nguyen
Keyword(s):  
Collocation Methods ◽  
Spline Collocation ◽  
Quadratic Spline ◽  
One Step ◽  
Quadratic Spline Collocation

A quadratic spline collocation method for the Dirichlet biharmonic problem

2019 ◽  
Vol 83 (1) ◽  
pp. 165-199 ◽  
Author(s):  
Bernard Bialecki ◽  
Graeme Fairweather ◽  
Andreas Karageorghis ◽  
Jonathan Maack
Keyword(s):  
Collocation Method ◽  
Spline Collocation ◽  
Quadratic Spline ◽  
Spline Collocation Method ◽  
Biharmonic Problem ◽  
Quadratic Spline Collocation
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