Stability analysis of the Eulerian-Lagrangian finite volume methods for nonlinear hyperbolic equations in one space dimension

TL;DR

This paper introduces a novel Eulerian-Lagrangian finite volume method for nonlinear hyperbolic equations in one dimension, improving stability and allowing larger time steps while preserving key properties like TVD and maximum principle.

Contribution

It develops a new ELFV scheme with adaptive space-time partitioning and troubled cell detection, enabling larger stable time steps compared to classical methods.

Findings

The scheme is total-variation-diminishing.

It preserves the maximum principle.

It allows at least twice larger time steps.

Abstract

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.