A high-order artificial compressibility method based on Taylor series time-stepping for variable density flow

TL;DR

This paper presents a fourth-order finite element method using Taylor series time-stepping for high-accuracy simulation of variable density incompressible flows, validated through numerical experiments.

Contribution

It introduces a novel fourth-order implicit time-stepping scheme based on Taylor series for variable density flow, with detailed algorithms and validation.

Findings

Achieves fourth-order accuracy for smooth problems

Fails for problems with loss of regularity like Rayleigh-Taylor instability

Provides detailed algorithms for time derivative approximation

Abstract

In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis functions in space. Taylor series time-stepping relies on time derivative correction terms to achieve high-order accuracy. We provide detailed algorithms to approximate the time derivatives of the variable density Navier-Stokes equations. Numerical validations confirm a fourth-order accuracy for smooth problems. We also numerically illustrate that the Taylor series method is unsuitable for problems where regularity is lost by solving the 2D Rayleigh-Taylor instability problem.