[Paper Review] Carbuncles as self-similar entropy solutions

Volker Elling|ArXiv.org|Sep 24, 2006

Computational Fluid Dynamics and Aerodynamics12 references3 citations

TL;DR

This paper proposes that carbuncles—numerical instabilities in shock wave simulations—are not mere artifacts but self-similar entropy solutions of the compressible Euler equations. Using similarity coordinates and a filament trigger mechanism, it demonstrates that carbuncles emerge as stable, self-similar structures across multiple numerical schemes, providing strong evidence they are non-physical but mathematically valid entropy solutions, challenging the assumption that they are purely numerical defects.

ABSTRACT

Numerical approximations of shock waves sometimes suffer from instabilities called carbuncles. Techniques for suppressing carbuncles are trial-and-error and lack in reliability and generality, partly because theoretical knowledge about carbuncles is equally unsatisfactory. It is not known which numerical schemes are affected in which circumstances, what causes carbuncles to appear and whether carbuncles are purely numerical artifacts or rather features of a continuum equation or model. This work presents evidence towards the latter: it is conjectured that carbuncles are a special class of non-physical entropy solutions. Using a new technique for triggering a single carbuncle, their structure is computed in detail in similarity coordinates.

Motivation & Objective

• To resolve the long-standing ambiguity about whether carbuncles are numerical artifacts or physical solutions by investigating their mathematical structure.
• To develop a reliable method for triggering and studying carbuncles in a controlled, reproducible manner across various numerical schemes.
• To determine whether carbuncles are part of a continuum family of entropy solutions, which would challenge the uniqueness and utility of the entropy condition.
• To enable precise computation of carbuncle structure through similarity coordinates, overcoming limitations of standard coordinate simulations.
• To lay the groundwork for theoretical proof of existence and control of carbuncle patterns, potentially leading to better numerical methods with minimal accuracy loss.

Proposed method

• Introduces a filament trigger mechanism—setting horizontal velocity to zero in a one-cell-high filament from the left boundary—to reliably initiate carbuncle formation in standard coordinates.
• Transforms the problem into similarity coordinates $(t, \vec{x}/t)$ to analyze self-similarity, revealing that carbuncles grow at constant velocity and settle into a stable, self-similar inner structure.
• Applies multiple numerical schemes (Godunov, Lax-Friedrichs, Osher-Solomon) and confirms carbuncles appear across all, indicating broad scheme dependence.
• Uses high-resolution simulations to examine the internal structure of carbuncles, particularly focusing on shock and contact discontinuity alignment and smearing.
• Analyzes the role of pre-shock Mach number (Mach 3.0 used), noting that carbuncles are more pronounced at higher Mach numbers and may be harder to trigger near Mach 1.4.
• Proposes that controlling free parameters like tip angle $\beta$ and tip location could enable convergence to a single carbuncle, supporting the hypothesis that they are entropy solutions.

Experimental results

Research questions

• RQ1 Are carbuncles numerical artifacts or do they represent a class of non-physical but mathematically valid entropy solutions of the Euler equations?
• RQ2 Can carbuncles be consistently triggered and studied across different numerical schemes using a unified method?
• RQ3 Do carbuncles exhibit self-similar structure, and if so, can this be used to compute their detailed internal dynamics with high accuracy?
• RQ4 Is there a continuous family of carbuncle patterns indexed by parameters such as tip angle $\beta$, and what would this imply for the entropy condition’s uniqueness?
• RQ5 Can the free parameters of carbuncles be controlled during simulation to enable convergence to a single solution, thereby strengthening evidence for their existence as entropy solutions?

Key findings

• Carbuncles are self-similar structures that grow at constant velocity and develop a stable internal configuration, confirmed via similarity coordinate analysis.
• The filament trigger mechanism reliably induces carbuncles across multiple schemes (Godunov, Lax-Friedrichs, Osher-Solomon), indicating broad applicability and not scheme-specific artifacts.
• Carbuncles persist even with high numerical viscosity away from the tip, suggesting the unphysical behavior is localized at the tip, not due to numerical diffusion.
• The internal structure of carbuncles is complex and smeared at coarse resolution, particularly for contact discontinuities not aligned with the grid, indicating need for finer grids.
• Carbuncles are observed to form at Mach numbers as low as 1.4 in standard coordinates, though triggering is difficult near this threshold, suggesting a possible threshold or mechanism limitation.
• Evidence supports that carbuncles are entropy solutions rather than non-entropy or purely numerical phenomena, with potential implications for the role of the entropy condition in hyperbolic conservation laws.