An Explicit Fourth-Order Hybrid-Variable Method for Euler Equations with A Residual-Consistent Viscosity

TL;DR

This paper introduces a fourth-order hybrid-variable method for Euler equations that combines superconvergence with residual-consistent viscosity, achieving high accuracy and stability in capturing discontinuities in 1D and 2D problems.

Contribution

The paper develops a novel fourth-order hybrid-variable method utilizing superconvergence and residual-consistent viscosity for improved Euler equation simulations.

Findings

Achieves fourth-order accuracy in Euler equations.

Effectively captures discontinuities with residual-consistent viscosity.

Demonstrates stability and high performance in 1D and 2D tests.

Abstract

In this paper we present a formally fourth-order accurate hybrid-variable method for the Euler equations in the context of method of lines. The hybrid-variable (HV) method seeks numerical approximations to both cell-averages and nodal solutions and evolves them in time simultaneously; and it is proved in previous work that these methods are inherent superconvergent. Taking advantage of the superconvergence, the method is built on a third-order discrete differential operator, which approximates the first spatial derivative at each grid point, only using the information in the two neighboring cells. Stability and accuracy analyses are conducted in the one-dimensional case for the linear advection equation; whereas extension to nonlinear systems including the Euler equations is achieved using characteristic decomposition and the incorporation of a residual-consistent viscosity to capture strong discontinuities. Extensive numerical tests are presented to assess the numerical performance of the method for both 1D and 2D problems.