Year: 2025
Type: article
Abstract: 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.
DOI: 10.5802/smai-jcm.135
Authors: Thierry Gallouët, Olivier Hurisse
Institution: Électricité de France (France)
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4407626092