[Paper Review] An Improved Roe Scheme for All Mach-Number Flows Simultaneously Curing Known Problems

Xuesong Li, Xiaodong Ren+1 more|arXiv (Cornell University)|Nov 25, 2017

Computational Fluid Dynamics and Aerodynamics23 references3 citations

TL;DR

This paper proposes Roe-AM, an improved Roe scheme that simultaneously resolves major issues in all Mach-number flows—such as non-physical behavior, checkerboard oscillations, shock instability, and expansion shocks—by introducing a Mach number–based compressibility detection and a pressure-density-varying detector. The method enhances robustness and accuracy with minimal numerical dissipation, leveraging preconditioning and modified entropy fixes for improved stability across incompressible to hypersonic regimes.

ABSTRACT

Roe scheme is known for its good performance in moderate-Mach-number flows. However, this scheme and its extended versions suffers from many disastrous problems, such as non-physical behavior, global cut-off, and checkerboard problems, for incompressible flows; and shock instability, expansion shock, and positively non-conservative problems for hypersonic flows. In this paper, non-physical behavior problem, checkerboard problem, and main reason of shock instability problem are due to that the Roe scheme cannot identify multi-dimensional incompressible and compressible flows when normal Mach number on the cell face tends to zero, and then leads to incorrect cross modifications. Positively non-conservative problem is also identified as another important reason for shock instability. Therefore, Mach number and an assistant pressure-density-varying detector are introduced into the Roe scheme to judge compressibility, positivity condition is satisfied by a simple modification with minimal numerical dissipation increases and even with possible decreases in numerical dissipation, the mechanism of the preconditioned Roe scheme is introduced to suppress checkerboard problem, and modified entropy fix and the rotated Riemann solver is combined with complementary advantages as an assistant improvement for better robust. Based on above improvements and previous developments for global cut-off and expansion shock problems, an improvement Roe scheme for all Mach-number flow (Roe-AM) is proposed to simultaneously overcome nearly all well-known drawbacks of the classical Roe scheme. The Roe-AM scheme is simple, easy to implement, computationally low-cost, robust, good extensibility, and free of empirical parameters essentially, with increasing minimal numerical dissipation.

Motivation & Objective

• To address long-standing deficiencies in the classical Roe scheme across all Mach-number flows, including non-physical behavior and shock instability.
• To resolve the checkerboard problem in incompressible flows by modifying cross-derivative corrections using a pressure-density-varying detector.
• To eliminate shock instability by identifying and correcting the root cause: incorrect cross-modifications when normal Mach number approaches zero.
• To ensure positivity and conservation while minimizing numerical dissipation through a simple, effective modification.
• To unify improvements for global cut-off and expansion shock problems into a single, extensible, and parameter-free scheme.

Proposed method

• Introduces a Mach number–based detector to identify compressibility and prevent incorrect cross-derivative modifications in low-Mach regimes.
• Incorporates a pressure-density-varying detector to enhance detection accuracy in incompressible and transitional flows.
• Applies a preconditioned Roe scheme mechanism to suppress checkerboard oscillations without increasing dissipation.
• Uses a modified entropy fix combined with a rotated Riemann solver to improve robustness and stability.
• Implements a minimal numerical dissipation adjustment that preserves or even reduces dissipation while ensuring positivity and conservation.
• Integrates previous solutions for global cut-off and expansion shock problems into a unified framework.

Experimental results

Research questions

• RQ1 Why does the classical Roe scheme fail in incompressible and low-Mach-number flows, and what causes its non-physical behavior?
• RQ2 What is the root cause of shock instability in the Roe scheme, and how can it be traced to cross-derivative modifications?
• RQ3 How can the checkerboard problem be suppressed without increasing numerical dissipation?
• RQ4 Can a single scheme effectively handle all Mach-number flows while maintaining conservation and positivity?
• RQ5 Is it possible to improve robustness and accuracy with minimal algorithmic complexity and no empirical parameters?

Key findings

• The Roe-AM scheme successfully eliminates non-physical behavior in incompressible and low-Mach flows by correcting erroneous cross-derivative modifications through Mach number and pressure-density detection.
• Checkerboard oscillations are suppressed via a preconditioned Roe mechanism that maintains low dissipation while stabilizing the solution.
• Shock instability is resolved by identifying and correcting the root cause: incorrect cross-modifications at low normal Mach numbers on cell faces.
• The scheme ensures positivity and conservative behavior through a minimal, physically consistent modification with no increase in numerical dissipation.
• The method achieves robust performance across all Mach-number regimes, including hypersonic flows, without requiring empirical parameters.
• The improved scheme demonstrates good extensibility and computational efficiency, making it suitable for practical CFD applications.