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Boundary Variation Diminishing (BVD) reconstruction: a new approach to improve Godunov scheme
Ziyao Sun, Satoshi Inaba, Feng Xiao
TL;DR
Reducing differences between reconstructed boundary states has been pursued for smooth solutions, but not developed as a scheme-design guideline, while higher-order polynomials do not necessarily reduce boundary jumps for discontinuous solutions. BVD adaptively selects reconstructions to minimize interface jumps; reported results include improved contact-discontinuity resolution and less oscillatory solutions without noticeable dissipation increase over WENO-only computation.
Problem
Reducing differences between reconstructed boundary states had not, to the authors’ knowledge, been explored as a design guideline, and higher-order polynomials do not necessarily reduce boundary jumps for discontinuous solutions.
Source: §1 · p. 2, §2.1 · p. 3Method
BVD adaptively chooses interpolation functions to minimize the jump between left- and right-side values at cell interfaces.
Source: §2.1 · p. 3Results
THINC significantly improves contact-discontinuity resolution, while BVD produces less oscillatory solutions without noticeable numerical-dissipation increase versus WENO-only computation.
Source: §2.2.2 · p. 4Takeaways & Limitations
BVD offers a reconstruction-design principle consistent with the smooth-solution boundary-jump observation and applicable to discontinuities.
Source: §1 · p. 2For discontinuous solutions, pursuing higher-order polynomials does not necessarily reduce reconstructed boundary jumps.
Source: §2.1 · p. 3
Abstract
from arXiv · showhide
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.
1. Introduction
Godunov schemes reconstruct cell-boundary states before computing numerical fluxes, and high-order methods have sought accuracy on smooth solutions while suppressing oscillations near discontinuities. The paper proposes boundary variation diminishing (BVD) reconstruction as a design guideline, building on the underexplored observation that reducing boundary-state differences may guide new schemes.
1. Introduction
- Godunov schemes reconstruct the physical fields at cell boundaries, then compute numerical fluxes to update cell-integrated values.The original scheme uses piecewise-constant reconstruction and exact Riemann solutions for boundary fluxes.Source: §1 · p. 1
- High-order polynomials improve smooth-solution accuracy, while nonlinear numerical dissipation is used to suppress spurious oscillations near discontinuities.The discussion situates these approaches against the Godunov barrier for monotonic linear schemes.Source: §1 · p. 1
- WENO provides a framework for high accuracy on smooth solutions and stabilization when discontinuities are present.The passage identifies improved smoothness indicators and nonlinear weights as one direction of subsequent WENO work.Source: §1 · p. 2
- Reducing differences between reconstructed left- and right-side values had been pursued for smooth solutions but, to the authors’ knowledge, was not explored as a scheme-design guideline.This observation motivates the proposed approach.Source: §1 · p. 2
- BVD reconstruction is proposed as a guideline for designing high-resolution schemes, demonstrated using 5th-order WENO and THINC.The guideline also provides a principle for devising reconstructions for discontinuities.Source: §1 · p. 2
2. Numerical algorithm
The algorithm uses a finite-volume framework and adaptively selects reconstructions to reduce jumps between left- and right-side values at cell interfaces. It combines fifth-order WENO and THINC candidates to address smooth and discontinuous solutions.
2. Numerical algorithm
- The finite-volume algorithm updates cell averages using Riemann fluxes computed from reconstructed interface values.The mesh is assumed uniform for simplicity.Source: §2 · p. 2
2.1. Boundary variation diminishing (BVD) reconstruction
- BVD adaptively chooses between candidate interpolants to minimize the jump between left- and right-side values at each cell interface.Its candidates pair a higher-order polynomial interpolant with a potentially lower-order, more monotone reconstruction that can represent a jump within a cell.Source: §2.1 · p. 3
- For smooth solutions, BVD naturally selects the highest possible interpolation, while higher-order polynomials do not necessarily reduce reconstructed boundary jumps for discontinuous solutions.For discontinuities, the authors prefer THINC to a polynomial because it can represent a jump within the cell.Source: §2.1 · p. 3
2.2. Two candidate reconstruction schemes
- The method uses fifth-order WENO and THINC as its two reconstruction candidates to resolve both smooth and discontinuous solutions.WENO uses nonlinear weights based on solution smoothness indicators.Source: §2.2 · p. 4, §2.2.1 · p. 4, §2.2.2 · p. 4
2.2.1. The 5th-order WENO reconstruction
- WENO computes interface values from third-order polynomials built on three shifted four-cell stencils.The nonlinear weights follow the WENO-Z scheme.Source: §2.2.1 · p. 4
2.2.2. The THINC reconstruction
- THINC uses a monotone hyperbolic-tangent sigmoid suited to step-like discontinuities, and its reconstruction significantly improves contact-discontinuity resolution.The jump-thickness parameter β is set to 1.6 for all numerical tests.Source: §2.2.2 · p. 4
- With piecewise-constant or MUSCL candidates for the second interpolant, BVD yields less oscillation without noticeable added numerical dissipation compared with WENO alone.The paper also notes that a total-boundary-variation criterion gives similar results but is more algorithmically complicated.Source: §2.1 · p. 3, §2.2.2 · p. 4
2.3. Some remarks on implementation
- For Euler equations, BVD minimizes reconstruction jumps separately in density, velocity, and pressure, uses Roe’s Riemann solver, and advances with five-stage fourth-order SSP Runge–Kutta at CFL 0.4.The same WENO and THINC reconstructions are used in the BVD procedure.Source: §2.3 · p. 5
3. Numerical results
The numerical-results section presents benchmark tests of the new algorithm. For smooth 1D linear advection, BVD-WENO-THINC matches WENO-Z errors across grid resolutions, recovering fifth-order WENO behavior.
3. Numerical results
- The section presents benchmark tests of the new algorithm, with detailed experiment setups referred to.Source: §3 · p. 5
3.1.1. Linear advection equation
- For smooth 1D linear advection, BVD-WENO-THINC has errors identical to WENO-Z across grid resolutions, recovering fifth-order WENO behavior.Table 1 reports numerical errors and convergence rates at t = 2.0.Source: §3.1.1 · p. 5, Table 1 · p. 5
- The accuracy test advects an initially smooth sine profile for one period and measures L1 and L∞ errors across grid resolutions.Source: §3.1.1 · p. 5
N BVD-WENO-THINC scheme WENO-Z scheme
The BVD-WENO-THINC scheme is reported to improve results across scalar and Euler conservation laws, with tests showing resolution of smooth and discontinuous features. The section also presents BVD as a reconstruction guideline that can use candidates beyond WENO-Z and THINC.
N BVD-WENO-THINC scheme WENO-Z scheme
- On Jiang and Shu’s mixed smooth/discontinuous test, BVD-WENO-THINC resolves the sharp discontinuity while retaining good resolution of the smooth profile.Source: N BVD-WENO-THINC scheme WENO-Z scheme · p. 6, Figure 1 · p. 6
3.1.2. Inviscid Burgers equation
- The sine-wave Burgers test is nearly fifth-order and resolves its later shock within two cells; the shock/rarefaction test recovers both features accurately.Source: §3.1.2 · pp. 6–7, Figure 2 · p. 8, Figure 3 · p. 8
3.1.3. Buckley Leverett equation
- For the Buckley–Leverett mixture of shock, rarefaction, and contact discontinuity, the results are reported as good quality on a low-resolution grid.Source: §3.1.3 · p. 7, Figure 4 · p. 9
3.2. 1D Euler equations
- The Sod and Lax tests capture sharp contacts and shocks while maintaining smooth-profile accuracy.Source: §3.2 · p. 7, Figure 5 · p. 9, Figure 6 · p. 10
- In the interacting-blast test, the left contact is resolved within fewer than four cells; shock-turbulence results reduce density-perturbation dissipation and remove spurious shock oscillation.Source: §3.2 · pp. 7, 11, Figure 7 · p. 10, Figure 8 · p. 11
3.3. 2D Euler equations
- The double Mach reflection and Mach 3 step results are reported as comparable in quality to DG models using even more degrees of freedom.Source: §3.3 · p. 11, Figure 9 · p. 12, Figure 10 · p. 13
3.4. Summary
- Across scalar and Euler laws, BVD-WENO-THINC substantially improves solution quality; BVD minimizes interface jumps, favoring high-order interpolation for smooth profiles and alternate reconstructions at discontinuities.The authors describe BVD as extensible to other candidate reconstructions.Source: §3.4 · pp. 11, 14