[Paper Review] A new shock-capturing scheme for stiff detonation waves problems

Xi Deng, Honghui Teng+2 more|arXiv (Cornell University)|Aug 3, 2017

Computational Fluid Dynamics and Aerodynamics18 references3 citations

TL;DR

This paper proposes the MUSCL-THINC-BVD scheme, a novel shock-capturing method that combines MUSCL and THINC reconstruction with a boundary variation diminishing (BVD) algorithm to drastically reduce numerical dissipation around discontinuities. By minimizing smearing of the detonation front, the scheme captures sharp, physically accurate detonation waves even on coarse meshes, outperforming high-order WENO schemes that produce spurious waves due to excessive dissipation.

ABSTRACT

A new approach to prevent spurious behavior caused by conventional shock-capturing schemes when solving stiff detonation waves problems is introduced in the present work. Due to smearing of discontinuous solution by the excessive numerical dissipation, conventional shock-capturing schemes have difficulties to obtain the correct location of detonation front without enough grids resolution. To overcome the excessive numerical errors around discontinuities by traditional discretized schemes used in non-reacting high speed compressible flow, we introduce a new shock-capturing scheme in which besides linear function constructed by MUSCL (Monotone Upstream-centered Schemes for Conservation Law) scheme, a step like THINC (Tangent of Hyperbola for INterface Capturing) function is also employed as another candidate in the reconstruction process. The final reconstruction function is determined by boundary variation diminishing (BVD) algorithm by which numerical dissipation around discontinuities can be reduced significantly. The new resulted shock-capturing scheme is named MUSCL-THINC-BVD. One- and two-dimensional comparative numerical tests about stiff detonation waves problems are conducted with the 5th order WENO (Weighted Essentially Non-Oscillatory) and MUSCL-THINC-BVD scheme respectively, which show MUSCL-THINC-BVD scheme can capture the correct position of detonation waves with improved resolution while WENO scheme, in spite of higher order, produces spurious waves. Compared with other existing methods which involves extra treatments by accepting the smeared out discontinuities profiles, the current method obtain the correct but also sharp detonation front by fundamentally reducing numerical dissipation errors from shock-capturing schemes. Thus the proposed approach is an effective but simple method to solve stiff detonation problems.

Motivation & Objective

• To address the spurious wave generation and incorrect detonation front propagation caused by excessive numerical dissipation in conventional shock-capturing schemes for stiff detonation problems.
• To overcome the limitations of high-order schemes like WENO, which produce non-physical oscillations despite high accuracy, due to smearing of discontinuities.
• To develop a method that fundamentally reduces numerical dissipation errors around shocks rather than relying on post-processing corrections or artificial treatments.
• To enable accurate simulation of detonation waves with minimal grid resolution by improving the reconstruction quality near discontinuities.
• To provide a simple yet effective alternative to complex front-tracking or adaptive mesh refinement techniques for stiff reactive flows.

Proposed method

• The method employs a hybrid reconstruction strategy using both a linear MUSCL function and a step-like THINC function as candidate reconstructions in each cell.
• The BVD (Boundary Variation Diminishing) algorithm selects the optimal reconstruction by minimizing total variation, effectively suppressing spurious oscillations and reducing numerical dissipation.
• The scheme is integrated into a finite volume framework for solving the reactive Euler equations with a stiff source term representing chemical reactions.
• The reconstruction is applied to conserved variables, and the resulting scheme is termed MUSCL-THINC-BVD to reflect its hybrid and dissipative-control nature.
• The method avoids external treatments such as temperature extrapolation or front tracking by inherently preserving sharp discontinuities through reduced dissipation.
• The approach is validated using 1D and 2D detonation problems with comparisons to 5th-order WENO and reference solutions on fine grids.

Experimental results

Research questions

• RQ1 Can a shock-capturing scheme reduce numerical dissipation around discontinuities in stiff detonation problems without relying on post-processing corrections?
• RQ2 Why do high-order schemes like WENO still produce spurious waves in under-resolved simulations of detonation waves?
• RQ3 How does the combination of MUSCL and THINC reconstructions with BVD selection improve resolution of sharp detonation fronts?
• RQ4 Can the MUSCL-THINC-BVD scheme achieve accurate detonation front location on coarse meshes where conventional schemes fail?
• RQ5 Is the proposed method effective for both stiff and non-stiff reactive flows without requiring problem-specific tuning?

Key findings

• The MUSCL-THINC-BVD scheme successfully captures the correct detonation front location on coarse meshes, while the 5th-order WENO scheme produces spurious waves due to excessive numerical dissipation.
• In the 1D interaction problem, MUSCL-THINC-BVD resolves the detonation front sharply without non-physical oscillations, whereas WENO generates spurious waves even with high-order accuracy.
• For the 2D detonation problem, MUSCL-THINC-BVD resolves the cellular structure and vortices behind the detonation wave accurately on a 400×80 grid, while WENO smears the reaction zone and introduces spurious waves.
• The scheme maintains accuracy and stability without requiring additional steps such as shock location detection or temperature extrapolation, unlike other advanced methods.
• The BVD algorithm effectively suppresses numerical oscillations and reduces dissipation, enabling the scheme to preserve sharp discontinuities even when the mesh is under-resolved.
• The method achieves improved resolution quality over standard shock-capturing schemes by fundamentally reducing dissipation errors at the discontinuity, rather than correcting them a posteriori.