Document Type

Article

Publication Date

2026

DOI

10.1007/s11075-026-02434-4

Publication Title

Numerical Algorithms

Volume

Advance online publication

Pages

22 pp.

Abstract

We propose a new method for parallelization of the first-order backward difference discretization (BDF1) of the first-order time derivative in nonlinear partial differential equations, such as conservation law 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 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 decoupled analytically by using the block Gaussian elimination so that each decoupled equation is associated with the corresponding time level. All product matrices on the left- and right-hand sides of this decoupled system of equations are calculated in parallel, such that their overall computational cost is linear in the number of spatial degrees of freedom. This allows for an efficient parallel-in-time implementation of the implicit BDF1 discretization for nonlinear differential equations of hyperbolic, parabolic or mixed type. Furthermore, the proposed parallel-in-time method preserves the quadratic rate of convergence of the Newton method of the sequential BDF1 scheme. 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 heat and Burgers equations with both smooth and discontinuous solutions.

Rights

© 2026 The Authors.

This article is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Data Availability

Article states: "The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request."

Original Publication Citation

Yamaleev, N. K., & Paudel, S. (2026). A new parallel-in-time direct inverse method for nonlinear differential equations. Numerical Algorithms. Advance online publication. https://doi.org/10.1007/s11075-026-02434-4

Download

Share

COinS