[Paper Review] Sharpening methods for finite volume schemes

Bruno Després, Samuel Kokh+1 more|arXiv (Cornell University)|Aug 9, 2016

Computational Fluid Dynamics and Aerodynamics60 references3 citations

TL;DR

This paper presents a comprehensive review of sharpening methods for finite volume schemes, focusing on high-order and nonlinear techniques to maintain sharp discontinuities in advection and hyperbolic systems. It details anti-diffusion, limited downwind reconstruction, Glimm's scheme, level set methods, and VOF-type interface tracking, demonstrating their ability to preserve contact discontinuities and material interfaces with minimal smearing under conservative, high-resolution schemes.

ABSTRACT

We review sharpening methods for finite volume schemes, with an emphasis on the basic structure of sharpening methods. It covers high order methods and non linear techniques for linear advection, Glimm's method, anti-diffusion techniques, the interaction of these techniques with the PDE structures. Additional approaches like level sets, interface reconstruction and Vofire are also discussed. We also present the algorithmic structure of the downwind method for a simple two components problem.

Motivation & Objective

• To analyze and systematize sharpening techniques that maintain sharp profiles in finite volume schemes for hyperbolic conservation laws.
• To address the challenge of numerical smearing of contact discontinuities and material interfaces in compressible flow simulations.
• To evaluate the effectiveness of high-order, nonlinear, and anti-diffusive methods in preserving interface sharpness without compromising conservation.
• To examine the algorithmic and mathematical foundations of interface reconstruction, including limited downwind and level set-based approaches.
• To clarify the interplay between sharpening mechanisms and underlying PDE structures, especially in multidimensional and nonlinear systems.

Proposed method

• Uses linear Strang’s stencils and compact finite difference formulations to derive high-order finite volume schemes with controlled dissipation.
• Applies anti-diffusion techniques via limited downwind reconstruction to counteract numerical diffusion in interface tracking.
• Employs the limited downwind strategy within bounds derived from local cell values and flux directions to ensure stability and sharpness.
• Integrates Glimm’s random choice method for sharp resolution of contact discontinuities in nonlinear systems.
• Adapts VOF and level set methods for interface reconstruction, particularly in two-phase compressible flows with complex interface dynamics.
• Combines conservative finite volume discretization for conserved variables with non-conservative treatment for interface evolution to maintain sharp profiles.

Experimental results

Research questions

• RQ1 How can high-order finite volume schemes be designed to minimize smearing of discontinuities in advection-dominated problems?
• RQ2 What is the role of anti-diffusion and limited downwind reconstruction in preserving sharp interfaces without introducing oscillations?
• RQ3 How do Glimm’s scheme and other nonlinear methods contribute to the accurate tracking of contact discontinuities in hyperbolic systems?
• RQ4 In what ways do level set and VOF-type reconstruction techniques improve interface resolution in multi-material compressible flows?
• RQ5 What are the stability and CFL constraints when using cut-cell geometries in dynamic interface simulations?

Key findings

• High-order finite volume schemes, such as the third-order O3 scheme, can achieve sharpening effects despite the inherent challenges of discontinuous solutions due to their compact stencil and conservative structure.
• The limited downwind reconstruction method ensures that interface values remain within physically consistent bounds, preserving monotonicity and preventing unphysical oscillations.
• Anti-diffusion techniques, when properly bounded, significantly reduce numerical smearing of material interfaces while maintaining conservation of mass and momentum.
• Glimm’s method enables sharp resolution of contact discontinuities by using random Riemann solvers to capture complex wave interactions with minimal dissipation.
• The Vofire algorithm avoids severe CFL restrictions in cut-cell scenarios, unlike standard methods, due to its unique treatment of interface geometry.
• Interface reconstruction using level sets or VOF-type methods enables accurate tracking of moving fronts in two-phase compressible flows, especially when combined with conservative fluxes.