Front propagation is a challenge in numerical modeling, particularly for multi-fluid or multi-material systems requiring clear material separation. The GRU (Glimm Random Update) scheme, inspired by Glimm’s method, has been developed to handle sharp fronts on unstructured 2D/3D meshes. It achieves convergence rates numerically measured between $0.8$ and $0.9$, which are higher than those of classical first-order schemes, typically limited to $0.5$ in the presence of contact discontinuities. Previous work established convergence in probability for planar fronts with uniform velocity but relied on random sequences, whereas practical implementations use low-discrepancy sequences. This study aims to establish new theoretical convergence results for the GRU scheme for 1D non-uniform meshes and uniform front velocity conditions, extending beyond uniform meshes considered previously. More precisely, we prove convergence in probability of order close to one-half when using random sequences on non-uniform meshes, and exact first-order convergence in probability when using low-discrepancy sequences (deterministic) on uniform meshes. Partial results are also obtained for non-uniform meshes, which are of significant practical interest. These findings provide insights into the scheme’s behavior with both random and low-discrepancy sequences. Numerical tests are presented to illustrate the theoretical results, with applications to non-uniform meshes, highlighting the scheme’s practical relevance.
Keywords: Front propagation, convergence, stochastic, low-discrepancy sequences
Thierry Gallouët 1; Olivier Hurisse 2
@article{SMAI-JCM_2025__11__585_0,
author = {Thierry Gallou\"et and Olivier Hurisse},
title = {Some theoretical results for the convergence of a {Glimm-like} scheme.},
journal = {The SMAI Journal of computational mathematics},
pages = {585--606},
year = {2025},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {11},
doi = {10.5802/smai-jcm.135},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.135/}
}
TY - JOUR AU - Thierry Gallouët AU - Olivier Hurisse TI - Some theoretical results for the convergence of a Glimm-like scheme. JO - The SMAI Journal of computational mathematics PY - 2025 SP - 585 EP - 606 VL - 11 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.135/ DO - 10.5802/smai-jcm.135 LA - en ID - SMAI-JCM_2025__11__585_0 ER -
%0 Journal Article %A Thierry Gallouët %A Olivier Hurisse %T Some theoretical results for the convergence of a Glimm-like scheme. %J The SMAI Journal of computational mathematics %D 2025 %P 585-606 %V 11 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.135/ %R 10.5802/smai-jcm.135 %G en %F SMAI-JCM_2025__11__585_0
Thierry Gallouët; Olivier Hurisse. Some theoretical results for the convergence of a Glimm-like scheme.. The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 585-606. doi: 10.5802/smai-jcm.135
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