A hat function based positive contact discontinuity capturing Boltzmann scheme

1. Introduction

• The PVU framework separates fluid and peculiar velocities, but its original scheme is overly diffusive and cannot resolve stationary discontinuities.Singh and Raghurama Rao designed a flux whose diffusion switches off across stationary contacts and remains low but non-zero otherwise, after an earlier approach proved unstable for strong nonlinear discontinuities.Source: §1 · p. 2
• The modified PVU scheme uses a compactly supported hat function that preserves the Euler moments and simplifies analysis enough to permit a direct positivity proof.The paper also reports numerical results equivalent to those obtained with the Maxwellian.Source: §1 · p. 2

2. From kinetic theory to macroscopic fluid dynamics

• At equilibrium, the Maxwell distribution's moments recover the Euler equations, while the Maxwellian's infinite velocity-space support complicates numerical analysis.The kinetic model represents flow through a particle velocity distribution, and the Euler equations follow from moments of the Boltzmann equation.Source: §1 · p. 2, §2 · pp. 3–4

2.1. Peculiar velocity based Boltzmann equation

• The peculiar-velocity formulation separates advective and pressure fluxes, distinguishing fluid-velocity propagation from random molecular motion.Peculiar velocity is defined relative to the bulk fluid velocity.Source: §2.1 · p. 5

3.1. Moments of the one-dimensional Maxwell distribution function

• In one dimension, four independent moment relations suffice to recover the Euler equations, and the hat function is constructed to satisfy those constraints.Its derivation is presented as an alternative to Perthame's original derivation.Source: §3.1 · p. 7, §3.2 · pp. 7–9

3.3. Equivalent form of the Euler equations of gas dynamics in 1D

• For internal degrees of freedom, two compactly supported distribution functions are used, and their Boltzmann-like equations recover the Euler equations as a linear combination.This formulation addresses the unbounded internal-energy support of the earlier distribution approach; the alternative was introduced by Perthame.Source: §3.3 · pp. 9–10

4. The one-dimensional peculiar velocity based upwind Boltzmann scheme with a hat function

• The scheme evolves cellwise equilibrium distributions through streaming between cells and collision relaxation that preserves mass, momentum, and total energy.The resulting finite-volume update uses interface fluxes formed from advective and pressure contributions.Source: §4 · pp. 11–12

4.1. Positivity analysis in 1D

• The cited result establishes that the peculiar-velocity-based upwind Boltzmann scheme (42)–(45) is positive, meaning its density is nonnegative (ρ ≥ 0) and its pressure is nonnegative (p ≥ 0).Source: §4.1 · p. 13

4.2. Inability of the PVU Scheme to capture stationary contact discontinuities

• For a stationary contact with zero velocity, equal pressure, and unequal densities, exact discrete equilibrium requires every interface flux to have the same value.The section states this as a sufficient condition for discrete equilibrium.Source: §4.2 · p. 14

5. A contact discontinuity capturing PVU Boltzmann scheme

• The design targets numerical diffusion that vanishes as Mach number goes to zero and remains nonzero otherwise, without compromising the original scheme’s robustness.The scheme addresses stationary linear discontinuities and spurious pressure and velocity oscillations near isolated contacts.Source: §5 · p. 15

5.1. Numerical formulation

• The numerical formulation separates advective and pressure fluxes and requires the diffusion-control parameters to satisfy dynamical consistency.The flux form is introduced generally for the two parts.Source: §5.1 · p. 15

5.2. Positivity analysis

• Under the stated assumptions and CFL-like condition, the modified scheme preserves nonnegative density and pressure.The diffusion controls are constrained to support positivity while reducing peculiar-velocity diffusion.Source: §5.2 · p. 20

5.3. Contact-discontinuity constraints on the diffusion control parameter

• The peculiar-velocity diffusion control must vanish on the stationary contact manifold, where velocity is zero and left and right pressures match.The advective diffusive flux must also vanish at stationary contacts.Source: §5.3 · p. 22

5.4. Selection of diffusion control parameters

• The diffusion control is selected from quantities that vanish for stationary contacts, including pressure and velocity differences and average velocity.These are identified as natural candidates for the control parameter.Source: §5.4 · p. 23

5.4.1. Variant 1

• Variant 1 uses a contact-discontinuity sensor to define the peculiar-velocity diffusion coefficient.The sensor is dimensionless.Source: §5.4.1 · p. 23

5.4.2. Variant 2

• Variant 2 uses a hyperbolic-tangent form for the diffusion control and the same sensor as Variant 1.Source: §5.4.2 · p. 23

5.4.3. Variant 3

• The modified PVU scheme admits stationary contact discontinuities as discrete equilibria for the stated diffusion-control parameters.The equilibrium holds for all cells and time steps.Source: §5.4.3 · p. 24

6.1. Moments of the Maxwellian distribution function in 2D

• Only eight of the twelve 2D conserved-variable and flux moments are independent; satisfying them recovers the Euler equations for γ=5/3.Source: §6.1 · p. 28

6.2. A compactly supported hat function in two dimensions

• A product of hat functions in c1 and c2 replaces the two-dimensional Maxwellian while satisfying three nonzero independent moment relations.The hat's maximum and half-widths are set by enforcing those relations.Source: §6.2 · pp. 28–29

6.3. Equivalent form of the Euler equations of gas dynamics in 2D

• The compactly supported equilibrium moments recover the conserved-variable and flux vectors for monatomic gases, and the Euler equations follow in the equilibrium limit of two Boltzmann-like equations.Source: §6.3 · pp. 30, 32
• To account for molecular internal degrees of freedom, the formulation uses two compactly supported distribution functions for polyatomic gases.The resulting conserved-variable and flux vectors are expressed using these distributions.Source: §6.3 · p. 31

7.1. Numerical formulation

• The finite-volume update advances cellwise distributions through streaming and collision, using edge-normal fluxes split into advective and pressure parts.The resulting macroscopic update is a first-order finite-volume scheme for the two-dimensional Euler equations.Source: §7.1 · pp. 32–34

7.2. Positivity analysis in two dimensions

• Under the theorem's assumptions and constraints, the two-dimensional modified PVU Boltzmann scheme preserves positive density and pressure.Source: §7.2 · p. 35

7.3. Selection of the diffusion control parameters

• With interface states satisfying stationary-slip conditions, the selected diffusion parameters preserve the slip surface as a discrete equilibrium for every cell and time step.Source: §7.3 · p. 38

8. Extension to second-order accuracy in two-dimensions

• A MUSCL-like extension uses piecewise-linear reconstruction and interface hats, applies Venkatakrishnan then Zhang-Shu limiting, and uses SSPRK-3 time stepping.The two-stage limiter reduces spurious oscillations and enforces positivity at interfaces.Source: §8 · pp. 39–41

8.1. Positivity analysis of the second-order scheme

• If density and pressure start positive, the Euler-forward second-order-in-space modified PVU update preserves both under its CFL-like condition.Source: §8.1 · pp. 41, 43

9.1. Experimental Order of Convergence

• The second-order scheme’s EOC study reports L1 and L2 errors and EOC across grid sizes for the smooth density test at CFL 0.2.The density solution is sinusoidal, and the test evolves to t = 0.5 on grids from 10 to 1280 cells.Source: §9.1 · pp. 43–44, Table 1 · p. 44

9.2. 1D results and discussions

• Across the one-dimensional contact and shock tests, the modified scheme preserves stationary contacts; the moving-contact variant affects oscillations, and second order shows subtle post-shock oscillations.All proposed variants preserve the stationary contact as a discrete equilibrium; for the moving contact, only the choice in (118) exhibits O(10^-5) pressure and velocity oscillations. The slowly moving shock is oscillation-free at first order, while second order has O(10^-3) post-shock oscillations.Source: Figure 3 · p. 45, §9.2 · pp. 45–46, Figure 4 · p. 45
• In the modified Sod test neither order produces entropy-violating solutions; the scheme remains robust at low density and captures key features of three strong discontinuities at both orders.The low-density case has an internal-energy overshoot near the center, described as common in upwind schemes.Source: Figure 6 · p. 46, §9.2 · p. 47, Figure 7 · p. 47

9.3.1. Oblique shock reflection

• For oblique shock reflection, the second-order scheme sharply captures the shocks and reaches residual levels of at least O(10^-7) across three meshes, with a non-monotonic residual bump.The residual bump occurs around time 0.8; residual reduction with refinement is overall but not strictly monotonic.Source: Figure 9 · p. 48, §9.3.1 · pp. 48–49, Figure 10 · p. 49

9.3.2. Horizontal slip flow

• The first-order scheme captures the isolated, steady, grid-aligned horizontal slip surface without numerical diffusion.Source: §9.3.2 · p. 49, Figure 11 · p. 50

9.3.3. Positivity test case

• The pressure-contour test reinforces that the second-order scheme preserves positivity of pressure and density.Source: Figure 12 · p. 50, §9.3.3 · p. 51

9.3.4. Shock diffraction; 9.3.5. Supersonic flow over a forward-facing step

• The second-order diffraction result has no entropy-violating solutions; the paper also presents density contours for the forward-facing-step case at t = 4.Source: Figure 13 · p. 51, §9.3.4 · p. 52, Figure 14 · p. 52

9.3.6. Double mach reflection; 9.3.7. Hypersonic flow over a half cylinder

• The modified scheme avoids the unphysical kinked Mach stem and the carbuncle instability in the double-Mach-reflection and hypersonic half-cylinder tests.Source: Figure 15 · p. 53, §9.3.6 · p. 53, Figure 16 · p. 54

10. Conclusions

• The contact-discontinuity-capturing scheme uses a compactly supported hat function, and its second-order extension preserves the positivity property of the first-order formulation.A CFL-like condition is derived for selecting the time step in second-order simulations.Source: §10 · p. 54
• One- and two-dimensional compressible-flow problems demonstrate the modified PVU Boltzmann scheme’s robustness and positivity-preserving nature.Source: §10 · p. 55

CRediT author statement

• Balwinder Singh is credited with methodology, conceptualization, investigation, software, validation, formal analysis, and original-draft writing; S. V. Raghurama Rao with conceptualization, investigation, formal analysis, supervision, and review and editing.Source: CRediT author statement · p. 55

Declaration of competing interest

• The authors report no monetary interests or individual relationships that could influence the material presented.Source: Declaration of competing interest · p. 55

• The research presented in this article does not rely on any datasets.Source: Data availability · p. 55

Appendix A. Details of the 1D numerical flux

• The 1D numerical-flux appendix identifies 2 as the diffusive pressure flux and introduces its expression.Source: Appendix A · p. 56
• For the modified PVU Boltzmann scheme, PVU+ is introduced as the diffusive flux associated with numerical pressure flux.Source: Appendix A · p. 56

Appendix B. Details of the 2D numerical flux

• The 2D numerical-flux appendix identifies e as the diffusive pressure flux and introduces its expression.Source: Appendix B · p. 57