A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations

08/03/2021
by   Jun Liu, et al.
0

In this paper, we propose a direct parallel-in-time (PinT) algorithm for time-dependent problems with first- or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix V grows as n is increased, where n is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing V and V^-1, by which we prove that Cond_2(V)=𝒪(n^2). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
01/25/2022

Substitution Method for Fractional Differential Equations

Numerical solving differential equations with fractional derivatives req...
research
04/23/2020

Subdiffusion with Time-Dependent Coefficients: Improved Regularity and Second-Order Time Stepping

This article concerns second-order time discretization of subdiffusion e...
research
06/09/2020

A pseudo-spectral Strang splitting method for linear dispersive problems with transparent boundary conditions

The present work proposes a second-order time splitting scheme for a lin...
research
01/07/2022

A Direct Parallel-in-Time Quasi-Boundary Value Method for Inverse Space-Dependent Source Problems

Inverse source problems arise often in real-world applications, such as ...
research
05/17/2021

Full operator preconditioning and the accuracy of solving linear systems

Unless special conditions apply, the attempt to solve ill-conditioned sy...
research
12/08/2020

Numerical analysis of a wave equation for lossy media obeying a frequency power law

We study a wave equation with a nonlocal time fractional damping term th...
research
09/27/2019

Efficient computation of the density matrix with error control on distributed computer systems

The recursive polynomial expansion for construction of a density matrix ...

Please sign up or login with your details

Forgot password? Click here to reset