Issue |
ESAIM: M2AN
Volume 59, Number 4, July-August 2025
|
|
---|---|---|
Page(s) | 1831 - 1861 | |
DOI | https://doi.org/10.1051/m2an/2025044 | |
Published online | 04 July 2025 |
Stability analysis of the Eulerian–Lagrangian finite volume methods for nonlinear hyperbolic equations in one space dimension
1
Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA
2
Department of Mathematics, University of Delaware, Newark, DE 19716, USA
* Corresponding author: yyang7@mtu.edu
Received:
5
May
2024
Accepted:
30
May
2025
In this paper, we construct a novel Eulerian–Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine–Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still limited. To fix this gap, we detect effective troubled cells (ETCs) and carefully design the influence region of each ETC, within which the partitioned space-time regions are merged together to form a new one. Then with the new partition of the space-time domain, we theoretically prove that the proposed first-order scheme with Euler forward time discretization is total-variation-diminishing and maximum-principle-preserving with at least twice larger time step constraints than the classical first order Eulerian method for Burgers’ equation. Numerical experiments verify the optimality of the designed time step sizes.
Mathematics Subject Classification: 65M08 / 65M12
Key words: Finite volume method / Eulerian–Lagrangian / nonlinear hyperbolic equations / stability analysis / total variation diminishing / maximum-principle
© The authors. Published by EDP Sciences, SMAI 2025
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