Volume 66, pp. 83-107, 2026.

An adaptive two-level nonlinear elimination preconditioned space-time solver for hyperbolic PDEs with shocks

Chang-Wan Liang and Feng-Nan Hwang

Abstract

Fully coupled space–time methods for time-dependent partial differential equations (PDEs) have recently gained popularity because of their potential to exploit parallelization in the temporal domain, driven by advances in computing power. These algorithms require solving large, sparse nonlinear systems simultaneously, necessitating a robust and efficient nonlinear solver as a critical component of the entire solution algorithm. This paper studies some nonlinear preconditioned Newton algorithms for the space–time formulation of hyperbolic equations with shocks. In such cases, the classical inexact Newton method with backtracking (INB) suffers from long stagnation due to local strong nonlinearity. Nonlinear preconditioning, such as nonlinear elimination, has been shown to improve the robustness of INB for many types of PDEs; however, it is not effective for hyperbolic PDEs. To address this, we investigate a variant of nonlinear elimination preconditioners for hyperbolic PDEs, considering their characteristics to mitigate the associated difficulties. To further reduce the overhead of the subspace correction step, we consider a two-level version of the nonlinear elimination preconditioned inexact Newton method (NEPIN), where the levels correspond to successive nonlinear elimination steps. The present study focuses on the mathematical properties, convergence behavior, and serial performance of the proposed algorithms. We conducted a comparative performance study of two nonlinear preconditioned iterative algorithms, INB with adaptive nonlinear elimination (INB-ANE) and NEPIN, where nonlinear elimination techniques act as right or left nonlinear preconditioners, respectively, in conjunction with inexact Newton algorithms. We present numerical results using Riemann problems for Burgers' equation and the Buckley–Leverett equation. This indicates that NEPIN outperforms INB-ANE in identifying the correct shock location and introducing less interface pollution after subspace correction, prior to the global update. Additionally, the number of Newton iterations required to converge for NEPIN is almost independent of the time step and mesh size. Finally, a stiffness analysis is conducted to provide further insight into the effectiveness of NEPIN in a hyperbolic space–time formulation.

Full Text (PDF) [4.1 MB], BibTeX , DOI: 10.1553/etna_vol66s83

Key words

parallel-in-time algorithms, space–time methods, hyperbolic PDEs, shock waves, nonlinear elimination preconditioning, inexact Newton method, coarse-grid correction, multilevel algorithms

AMS subject classifications

65H10, 49M15

Links to the cited ETNA articles

[16]	Vol. 37 (2010), pp. 239-251 Feng-Nan Hwang, Hsin-Lun Lin, and Xiao-Chuan Cai: Two-level nonlinear elimination based preconditioners for inexact Newton methods with application in shocked duct flow calculation