Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A New Parallel-in-time Direct Inverse Method for Nonlinear Differential Equations

Nail K. Yamaleev, Subhash Paudel

TL;DR

This work introduces ParaDIn, a parallel-in-time method for nonlinear PDEs discretized with implicit BDF1, by solving the all-at-once space-time Newton system. The key idea is to transform the block-bidiagonal time-coupled system into a decoupled form through an inverse-like construction, enabling independent time-level solves while preserving the quadratic convergence of Newton iterations. The authors prove the equivalence of the decoupled and original systems and demonstrate nearly ideal speedups on up to 32 cores for 2-D nonlinear heat and Burgers equations, including cases with smooth and discontinuous solutions. Practical results show substantial time-parallel efficiency, with detailed analysis of computational cost, conditioning, and initialization strategies. The approach is compatible with standard spatial discretizations and holds promise for large-scale, time-parallel simulations of nonlinear PDEs.

Abstract

We present a new approach to parallelization of the first-order backward difference discretization (BDF1) of the time derivative in partial differential equations, such as the nonlinear heat and viscous Burgers equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order 3-point central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at each Newton iteration is block bidiagonal, which can be inverted directly in a blockwise manner, thus leading to a set of fully decoupled equations associated with each time level. This allows for an efficient parallel-in-time implementation of the implicit BDF1 discretization for nonlinear differential equations. The proposed parallel-in-time method preserves a quadratic rate of convergence of the Newton method of the sequential BDF1 scheme, so that the computational cost of solving each block matrix in parallel is nearly identical to that of the sequential counterpart at each time step. Numerical results show that the new parallel-in-time BDF1 scheme provides the speedup of up to 28 on 32 computing cores for the 2-D nonlinear partial differential equations with both smooth and discontinuous solutions.

A New Parallel-in-time Direct Inverse Method for Nonlinear Differential Equations

TL;DR

This work introduces ParaDIn, a parallel-in-time method for nonlinear PDEs discretized with implicit BDF1, by solving the all-at-once space-time Newton system. The key idea is to transform the block-bidiagonal time-coupled system into a decoupled form through an inverse-like construction, enabling independent time-level solves while preserving the quadratic convergence of Newton iterations. The authors prove the equivalence of the decoupled and original systems and demonstrate nearly ideal speedups on up to 32 cores for 2-D nonlinear heat and Burgers equations, including cases with smooth and discontinuous solutions. Practical results show substantial time-parallel efficiency, with detailed analysis of computational cost, conditioning, and initialization strategies. The approach is compatible with standard spatial discretizations and holds promise for large-scale, time-parallel simulations of nonlinear PDEs.

Abstract

We present a new approach to parallelization of the first-order backward difference discretization (BDF1) of the time derivative in partial differential equations, such as the nonlinear heat and viscous Burgers equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order 3-point central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at each Newton iteration is block bidiagonal, which can be inverted directly in a blockwise manner, thus leading to a set of fully decoupled equations associated with each time level. This allows for an efficient parallel-in-time implementation of the implicit BDF1 discretization for nonlinear differential equations. The proposed parallel-in-time method preserves a quadratic rate of convergence of the Newton method of the sequential BDF1 scheme, so that the computational cost of solving each block matrix in parallel is nearly identical to that of the sequential counterpart at each time step. Numerical results show that the new parallel-in-time BDF1 scheme provides the speedup of up to 28 on 32 computing cores for the 2-D nonlinear partial differential equations with both smooth and discontinuous solutions.
Paper Structure (11 sections, 3 theorems, 26 equations, 2 figures, 6 tables)

This paper contains 11 sections, 3 theorems, 26 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Governing equations and the sequential BDF1 scheme
  3. Global-in-time formulation of the BDF1 scheme
  4. New parallel-in-time BDF1 scheme
  5. Parallel algorithm for calculating left- and right-hand sides of Eq. (\ref{['eq:Dec']})
  6. Computational cost and speedup
  7. Influence of the condition number
  8. Numerical results
  9. 2-D nonlinear heat equation
  10. 2-D viscous Burgers equation
  11. Conclusions

Key Result

Theorem 1

If matrices $A_i$, $i=1,\dots,N_t$ in Eq. (eq:GlobalNewton) are nonsingular, i.e., $\det A_i \ne 0, \ \forall i$, then the following system of equations has a unique solution that is identical to that of Eq. (eq:GlobalNewton).

Figures (2)

  • Figure 1: Convergence histories of Newton iterations obtained with the parallel-in-time and sequential BDF1 schemes on the grid with $N_t=8$ and $N_x=N_y=64$ for the 2-D nonlinear heat equation.
  • Figure 2: Convergence histories of Newton iterations obtained with the parallel-in-time and sequential BDF1 schemes on the grid with $N_t=12$ and $N_x=N_y=63$ for the 2-D Burgers equation.

Theorems & Definitions (10)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Remark 3
  • Remark 4