Boundary Variation Diminishing (BVD) reconstruction: a new approach to improve Godunov scheme

Abstract: This paper presents a new approach, so-called boundary variation diminishing (BVD), for reconstructions that minimize the discontinuities (jumps) at cell interfaces in Godunov type schemes. It is motivated by the observation that diminishing the jump at the cell boundary might effectively reduce the dissipation in numerical flux. Different from the existing practices which seek high-order polynomials within mesh cells while assuming discontinuities being always at the cell interfaces, we proposed a new strategy that combines a high-order polynomial-based interpolation and a jump-like reconstruction that allows a discontinuity being partly represented within the mesh cell rather than at the interface. It is shown that new schemes of high fidelity for both continuous and discontinuous solutions can be devised by the BVD guideline with properly-chosen candidate reconstruction schemes. Excellent numerical results have been obtained for both scalar and Euler conservation laws with substantially improved solution quality in comparison with the existing methods. This work provides a simple and accurate alternative of great practical significance to the current Godunov paradigm which overly pursues the smoothness within mesh cell under the questionable premiss that discontinuities only appear at cell interfaces.

• The paper introduces Boundary Variation Diminishing (BVD) reconstruction, a novel method to improve Godunov schemes by reducing numerical dissipation at cell boundaries for better accuracy.
• BVD combines high-order polynomial interpolation with a discontinuity-flexible reconstruction strategy, specifically leveraging WENO and THINC schemes, to accurately handle both smooth regions and sharp discontinuities.
• Numerical results demonstrate that the BVD approach achieves superior accuracy and sharper resolution of discontinuities in benchmark tests compared to standard methods, with significant implications for computational fluid dynamics.

An Expert's Analysis of the BVD Reconstruction Approach for Godunov Schemes

The paper under discussion presents a novel reconstruction methodology termed Boundary Variation Diminishing (BVD), designed to enhance the performance of Godunov schemes in solving hyperbolic conservation laws, particularly the challenges posed by discontinuities. This analysis provides a detailed summary of the BVD approach, the numerical improvements it achieves, and its implications for computational fluid dynamics.

The Godunov scheme, known for its effectiveness in capturing shocks and discontinuities in fluid dynamics through a conservative finite-volume methodology, often faces efficiency limitations due to numerical dissipation linked to capturing discontinuities at cell boundaries. The primary innovation of the BVD approach lies in its different handling of discontinuities by reducing boundary variations at cell interfaces, thereby achieving higher fidelity in numerical solutions across both continuous and discontinuous regimes.

Key Contributions and Methodology

The paper identifies two primary procedural stages in Godunov-type schemes: reconstructing physical fields to ascertain interface states and evaluating boundary fluxes through Riemann solvers. The BVD approach specifically targets the reconstruction stage, diverging from traditional methods that predominantly employ high-order polynomial interpolations under the presumption of discontinuities strictly occurring at cell interfaces.

1. BVD Strategy: BVD combines high-order polynomial interpolation with a discontinuity-flexible reconstruction, allowing discontinuities to be represented within the cell rather than at its boundaries. The method effectively minimizes numerical dissipation by solving for reconstructions that diminish the jump magnitude at boundaries.
2. Implementation with WENO and THINC: For the practical implementation of BVD, the paper leverages a combination of the fifth-order WENO method and the THINC scheme, using BVD-guided selection to devise higher accuracy reconstructions. This dual reconstruction capability ensures that the scheme possesses robustness in handling both smooth variations and abrupt changes in the solution profile.

Numerical Results and Performance

The paper provides a series of benchmark tests demonstrating the superior performance of the BVD approach against standard Godunov methods. For scalar conservation laws and complex Euler equations, BVD achieves improved accuracy in both smooth contours and sharp discontinuities. For instance, in the resolution of shock tubes and interaction of blast waves, BVD displays pronounced sharpness at discontinuities compared to WENO alone, without introducing significant oscillations.

Implications and Future Directions

The BVD methodology introduces a flexible and efficient approach to resolving the Godunov paradigm's limitations. By reducing inaccuracies associated with boundary numerical dissipation, BVD has significant implications for computational fluid dynamics, particularly in simulations requiring fine resolution of structure within the fluid, such as in turbulence or shock-turbulence interactions.

Future work may explore extending the BVD approach to different discretization frameworks and integrating it with alternative reconstruction schemes to further maximize its potential in a broader array of high-resolution simulation scenarios. Moreover, integrating BVD with advanced Riemann solvers could further refine its applicability across more complex flow regimes.

Overall, the paper provides a substantial advancement in reducing dissipation in Godunov schemes through innovative boundary-aware reconstructions, contributing meaningfully to the computational techniques available for simulating hyperbolic conservation laws with improved accuracy and consistency.