# A Central Differential Flux with High-Order Dissipation for Robust Simulations of Transcritical Flows

## TL;DR

This paper introduces the CDHD numerical flux solver, combining central flux and high-order dissipation, to improve the robustness and accuracy of transcritical flow simulations while reducing energy errors.

## Contribution

The paper presents a novel primitive-variable flux solver with high-order dissipation that enhances stability and energy conservation in transcritical flow simulations.

## Key findings

- Reduces energy conservation error by two orders of magnitude.
- Effectively suppresses spurious pressure oscillations.
- Robustly handles smooth transcritical flows and shock waves.

## Abstract

The simulation of transcritical flows remains challenging due to strong thermodynamic nonlinearities that induce spurious pressure oscillations in conventional schemes.While primitive-variable formulations offer improved robustness under such conditions, they are always limited by energy conservation errors and the absence of systematic high-order treatments for numerical fluxes. In this paper, we introduce the Central Differential flux with High-Order Dissipation (CDHD), a novel numerical flux solver designed for primitive-variable discretization. This method combines a central flux for advection with a minimal, upwind-biased dissipation term to stabilize the simulation while maintaining formal accuracy. The dissipation term effectively suppresses oscillations and improves stability in transcritical flows. Compared to traditional primitive-variable approaches, CDHD reduces the energy conservation error in two order of magnitude. When incorporated into a hybrid framework with a conservative shock-capturing scheme, the method robustly handles both smooth transcritical phenomena and shock waves. Numerical tests validate the accuracy, stability, and energy-preserving capabilities of CDHD, demonstrating its potential as a reliable tool for complex real-gas flow simulations.