Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates

TL;DR

This paper introduces a novel finite difference method for 1D viscous isentropic flow, employing dual-grid upwind discretization for the momentum equation, and proves its convergence.

Contribution

It presents a new finite difference scheme with dual-grid upwind discretization for the momentum equation and establishes its convergence.

Findings

Method converges as discretization parameters tend to zero.

Adapts Lions and Feireisl's existence theory to numerical analysis.

Provides a rigorous proof of convergence for the proposed scheme.

Abstract

We construct a new finite difference method for the flow of ideal viscous isentropic gas in one spatial dimension. For the continuity equation, the method is a standard upwind discretization. For the momentum equation, the method is an uncommon upwind discretization, where the moment and the velocity are solved on dual grids. Our main result is convergence of the method as discretization parameters go to zero. Convergence is proved by adapting the mathematical existence theory of Lions and Feireisl to the numerical setting.