Pressure Jumped 21% in a Problem Where Nothing Should Happen — The Abgrall Criterion and the Nonconservative Term

The First Test Was a Problem Where Nothing Should Happen#

I picked the first verification case for a multicomponent compressible solver. Two gases meet across a single contact surface. Pressure is 100 kPa everywhere, velocity is 100 m/s everywhere. Only the density and the specific heat ratio of the two gases differ.

The exact solution is boring. The contact surface drifts to the right, and that is all — pressure and velocity stay uniform to the end. A coarse grid does not change that, and neither does a large time step. There is no wave in this problem to resolve.

After 20 steps, the pressure in the interface cell had dropped to 78.6 kPa. That is 21% off the uniform value. Velocity held at exactly 100 m/s, and mass and energy were conserved to machine precision. The flux was fine. What was wrong was the term standing next to the flux.

Let's produce that oscillation directly in the simulation below.

Both curves move the same conserved variables with the same upwind operator. The only difference is how the thermodynamic variable rides along. Pull gamma_2 toward 1.400 and the red curve settles onto the green one. Push it the other way, or raise the density ratio, and the red curve caves in at the two interface locations. Whether pressure stays uniform is decided not by the accuracy of the scheme but by this single choice.

What Exactly Went Wrong in This Calculation#

Pressure is not a conserved variable. It is recovered through the equation of state. When the specific heat ratio $\gamma$ varies in space,

Here $\rho E$ is total energy per unit volume, and $\Gamma$ is the thermodynamic variable that carries the specific heat ratio.

At uniform pressure and velocity the total energy is $\rho E = P\Gamma + \tfrac{1}{2}\rho u^{2}$. Upwind advection is linear, so the updated $\rho E$ equals $P \cdot (\Gamma$'s upwind advection$)$ plus the kinetic energy term. The new pressure then reduces to this.

$\mathcal{U}$ is that very upwind operator, the one actually applied to $\rho E$.

The condition boils down to a single fraction. Pressure stays put only if $\Gamma^{n+1}$ comes out equal to $\mathcal{U}(\Gamma)$. In other words, the way you move $\Gamma$ has to mesh algebraically with the way you move energy.

My first version of the code advected the mass fraction $Y$ in conservative form and recovered $\Gamma$ from a mixture rule. $\Gamma(Y)$ is linear in $Y$, but $Y^{n+1} = \mathcal{U}(\rho Y)/\mathcal{U}(\rho)$ is not $\mathcal{U}(Y)$ wherever the density varies. When the two fractions disagree, that discrepancy is exactly the pressure error.

The One-Line Requirement Abgrall Wrote Down in 1996#

The criterion Re and Abgrall cite while building their weakly compressible multicomponent model is a single sentence. "A two-phase flow that is uniform in pressure and velocity must remain uniform in the same variables as time advances." The paper calls it the pressure non-disturbance condition, or Abgrall's criterion.

What makes this sentence unusual is that it is not a demand on accuracy. First order or fifth order, it makes no difference. It is not a stability condition either. Lower the CFL number and the oscillation survives untouched. This is a demand for algebraic consistency between discretizations. Which operator you used in one equation determines which operators are permitted in the others.

The same situation shows up in the seven-equation model that gives each phase its own equations when the nonconservative interface term is chosen. There each phase keeps its own velocity and pressure; here the discretization of that term is chosen so uniform flow survives. Several plausible discretizations exist and only one of them does.

Two Discretizations Standing Side by Side on One Grid in Python#

I placed a single contact surface on a 100-cell periodic grid and ran both approaches for the same number of steps. The conserved variables $\rho$, $\rho u$, $\rho E$ are updated with the identical first-order upwind scheme in both. The only difference is one thermodynamic variable.

G1, G2 = 1.4, 1.667      # specific heat ratios of the two gases
P0, U0 = 1.0e5, 100.0    # uniform pressure [Pa], uniform velocity [m/s]
R1, R2 = 1.0, 0.125      # densities of the two gases [kg/m^3]
 
 
def gamma_var(y):
    """Mass fraction y -> 1/(gamma-1). A mixture rule linear in y."""
    return y / (G1 - 1.0) + (1.0 - y) / (G2 - 1.0)
 
 
def advect_upwind(q, lam):
    """First-order upwind advection for u>0. The left inflow cell is held fixed."""
    return [q[0]] + [q[j] - lam * (q[j] - q[j - 1]) for j in range(1, len(q))]
 
 
def initial_state(n):
    x = [(j + 0.5) / n for j in range(n)]
    y = [1.0 if xi < 0.3 else 0.0 for xi in x]
    rho = [R1 if xi < 0.3 else R2 for xi in x]
    return x, y, rho
 
 
def step_massfraction_closure(rho, mom, ene, ry, lam):
    """Advect rho*Y in conservative form and recover gamma from the mixture rule."""
    rho_n = advect_upwind(rho, lam)
    mom_n = advect_upwind(mom, lam)
    ene_n = advect_upwind(ene, lam)
    ry_n = advect_upwind(ry, lam)
    y_n = [ry_n[j] / rho_n[j] for j in range(len(rho_n))]
    p_n = [(ene_n[j] - 0.5 * mom_n[j] ** 2 / rho_n[j]) / gamma_var(y_n[j])
           for j in range(len(rho_n))]
    return rho_n, mom_n, ene_n, ry_n, p_n
 
 
def step_gammavar_transport(rho, mom, ene, gv, lam):
    """Carry 1/(gamma-1) in nonconservative (advective) form on the same upwind operator."""
    rho_n = advect_upwind(rho, lam)
    mom_n = advect_upwind(mom, lam)
    ene_n = advect_upwind(ene, lam)
    gv_n = advect_upwind(gv, lam)
    p_n = [(ene_n[j] - 0.5 * mom_n[j] ** 2 / rho_n[j]) / gv_n[j]
           for j in range(len(rho_n))]
    return rho_n, mom_n, ene_n, gv_n, p_n
 
 
def run_interface_advection(n=100, steps=60, cfl=0.5):
    x, y0, rho0 = initial_state(n)
    gv0 = [gamma_var(v) for v in y0]
 
    rho_a = list(rho0)
    mom_a = [r * U0 for r in rho0]
    ene_a = [P0 * gv0[j] + 0.5 * rho0[j] * U0 ** 2 for j in range(n)]
    ry_a = [rho0[j] * y0[j] for j in range(n)]
 
    rho_b, mom_b, ene_b = list(rho_a), list(mom_a), list(ene_a)
    gv_b = list(gv0)
 
    hist, p_a, p_b = [], None, None
    for k in range(1, steps + 1):
        rho_a, mom_a, ene_a, ry_a, p_a = step_massfraction_closure(
            rho_a, mom_a, ene_a, ry_a, cfl)
        rho_b, mom_b, ene_b, gv_b, p_b = step_gammavar_transport(
            rho_b, mom_b, ene_b, gv_b, cfl)
        if k % 20 == 0:
            ea = max(abs(v - P0) for v in p_a)
            eb = max(abs(v - P0) for v in p_b)
            eu = max(abs(mom_a[j] / rho_a[j] - U0) for j in range(n))
            hist.append((k, ea, eb, eu))
    return x, p_a, p_b, hist
 
 
if __name__ == "__main__":
    x, p_a, p_b, hist = run_interface_advection()
    print("step |  max|P-P0| mixrule |  max|P-P0| Gamma-adv |  max|u-U0| mixrule")
    for k, ea, eb, eu in hist:
        print("%4d | %16.2f | %19.2e | %16.3e" % (k, ea, eb, eu))
 
    j = max(range(len(p_a)), key=lambda i: abs(p_a[i] - P0))
    print("\nworst cell x=%.3f  P=%.1f Pa  (uniform value %.0f Pa)" % (x[j], p_a[j], P0))
    print("relative error:  mixrule %.2f%%   Gamma-adv %.1e%%"
          % (max(abs(v - P0) for v in p_a) / P0 * 100,
             max(abs(v - P0) for v in p_b) / P0 * 100))

step |  max|P-P0| mixrule |  max|P-P0| Gamma-adv |  max|u-U0| mixrule
  20 |         21433.25 |            2.91e-11 |        0.000e+00
  40 |         21587.18 |            4.37e-11 |        0.000e+00
  60 |         21442.28 |            4.37e-11 |        2.842e-14
 
worst cell x=0.635  P=78557.7 Pa  (uniform value 100000 Pa)
relative error:  mixrule 21.44%   Gamma-adv 4.4e-14%

Three lines are all you need to read. The mixture-rule side starts at 21.4 kPa and does not shrink as the steps pile up. The $\Gamma$-advection side sits at $10^{-11}$ Pa, a relative error of $10^{-14}$%. That is the floor of double precision.

The zero column for velocity matters too. The momentum equation was solved correctly from beginning to end. Double the grid resolution and 21% stays 21%. This is an error that does not converge away.

The peak error is 21,433 Pa at step 20, 21,587 Pa at step 40, and 21,442 Pa at step 60. Triple the step count and the error stays glued near 21.4 kPa. The worst cell is at x = 0.635, where the pressure is 78,558 Pa. That is 100,000 minus 21,442. The relative error of 21.44% is the same 21% and the same 78.6 kPa from the opening. The floor was already reached at step 20. After that, only the contact moves.

At CFL 0.5 each step shifts the interface by half a cell. Sixty steps is 30 cells, or $\Delta x = 0.30$ on a 100-cell grid. The contact started at x = 0.3, so after 60 steps it sits near x = 0.6. The worst cell at 0.635 is the smeared interface, not a new spike. The error changes address. It does not change size.

The $\Gamma$-advection relative error at step 60 is $4.4\times 10^{-14}$%. That is fifteen digits away from the mixture-rule 21.44%. The velocity error on the same step is $2.842\times 10^{-14}$ m/s. The density ratio $1/0.125 = 8$ enters the denominator $\mathcal{U}(\rho)$, so a $\Gamma$ rebuilt from $Y$ no longer matches the $\Gamma$ that rode with the energy. Specific-heat ratios 1.4 and 1.667 become $\Gamma = 1/(\gamma-1)$, that is 2.5 and about 1.5. The variable that has to travel linearly is this $\Gamma$, not $Y$.

How the Paper Derives $H_u$ — Fix the Scheme First, Then Match the Term#

The Baer–Nunziato-type model of Re and Abgrall (the seven-equation family, where each phase carries its own velocity and pressure) has a separate volume fraction equation. That equation is not in conservative form.

$\alpha_i$ is the volume fraction of phase $i$, and $u_I$ is the interface velocity. The paper leaves this term as a discrete operator named $H_u(\alpha_i, u_I)$ and does not assume its form. It derives it from the condition instead.

The mass equation $(\alpha_i\rho_i)$ is already fixed to the Rusanov flux. Feed it a state with uniform density and velocity and that flux factors into $\rho_i \times (\alpha_i$'s Rusanov flux$)$. For $\rho_i = (\alpha_i\rho_i)^{n+1}/\alpha_i^{n+1}$ to survive the update unchanged, the denominator $\alpha_i^{n+1}$ has to be updated with exactly the same flux difference as the numerator. So $H_u$ is not a choice — it is a consequence.

The first term is a central difference, the second is diffusion weighted by $|u_I|$. Their sum is exactly the Rusanov flux difference for $\alpha$. When $u_I > 0$, only $\alpha_j - \alpha_{j-1}$ remains. Pure upwinding.

The paper reuses the same $H_u$ in the pressure equation. It splits $u^{*}\partial P/\partial x$ into $\partial(Pu^{*})/\partial x - P\partial u^{*}/\partial x$, then puts the leftover nonconservative term on the same operator as the mass equation. Using a different discretization per equation would break the consistency just established.

Turning the Three-Point Stencil Coefficients by Hand#

Attach a weight $\theta$ to the diffusive one of the two terms in $H_u$ and a single dial sweeps between the two extremes. At $\theta = 1$ you get the paper's $H_u$; at $\theta = 0$ you get a pure central difference of $u_I \partial\alpha/\partial x$.

I checked it numerically as well. Uniform density of 850 kg/m³, a single interface, the same grid, with only $\alpha$ updated by the two operators.

RHO, UI, N, LAM = 850.0, 1.0, 80, 0.4   # uniform density [kg/m^3], interface velocity, cell count, u*dt/dx
 
 
def alpha_profile():
    """Volume fraction across an interface. Bridges 0.02 <-> 0.98 over three cells."""
    a = []
    for j in range(N):
        if j < 30:
            a.append(0.98)
        elif j < 33:
            a.append(0.98 - 0.32 * (j - 29))
        else:
            a.append(0.02)
    return a
 
 
def rusanov_flux(q, j, vel):
    """Rusanov numerical flux between cell j and j+1 (periodic boundary)."""
    ql, qr = q[j % N], q[(j + 1) % N]
    return 0.5 * (qr + ql) * vel - 0.5 * abs(vel) * (qr - ql)
 
 
def hu_upwind(a, j):
    """Nonconservative operator of Eq. (10) in the paper: Rusanov flux difference for alpha."""
    return rusanov_flux(a, j, UI) - rusanov_flux(a, j - 1, UI)
 
 
def hu_central(a, j):
    """Version that discretizes u_I * d(alpha)/dx directly with a central difference."""
    return 0.5 * UI * (a[(j + 1) % N] - a[(j - 1) % N])
 
 
def march(op, steps):
    """Advance alpha*rho with Rusanov and alpha with op, then inspect the recovered rho."""
    a = alpha_profile()
    ar = [RHO * v for v in a]
    for _ in range(steps):
        ar = [ar[j] - LAM * (rusanov_flux(ar, j, UI) - rusanov_flux(ar, j - 1, UI))
              for j in range(N)]
        a = [a[j] - LAM * op(a, j) for j in range(N)]
    return max(abs(ar[j] / a[j] - RHO) for j in range(N))
 
 
if __name__ == "__main__":
    print("steps |  upwind H_u [kg/m^3] |  centred [kg/m^3]")
    for s in (10, 40, 120):
        print("%5d | %20.2e | %17.4f" % (s, march(hu_upwind, s), march(hu_central, s)))
 
    a = alpha_profile()
    lhs = hu_upwind(a, 31)
    rhs = 0.5 * ((a[32] - a[30]) * UI - abs(UI) * (a[32] - 2 * a[31] + a[30]))
    print("\ncell 31:  flux difference %.6f   paper Eq.(10) %.6f   gap %.1e"
          % (lhs, rhs, abs(lhs - rhs)))

steps |  upwind H_u [kg/m^3] |  centred [kg/m^3]
   10 |             2.27e-13 |         2553.2027
   40 |             4.55e-13 |         6697.6224
  120 |             5.68e-13 |         1206.6015
 
cell 31:  flux difference -0.320000   paper Eq.(10) -0.320000   gap 0.0e+00

Turn the theta dial down from 1 and the downstream weight comes back, taking the recovered density off 850 kg/m³.

Flip u_I and $\theta = 1$ still holds 850. The central stencil does not follow the sign. The centred column in the output above is that departure.

The upwind $H_u$ stays at $10^{-13}$ kg/m³ even after 120 steps. The central difference drifts 2,553 kg/m³ away within 10 steps. The value coming back down to 1,207 at 120 steps is not recovery but divergence. It means $\alpha$ has wandered near zero and entered a regime where the division returns whatever it likes.

The last line confirms the derivation. The Rusanov flux difference for $\alpha$ and the closed form of Eq. (10) in the paper agree at cell 31 down to the last digit. The two expressions are algebraically the same thing.

What It Means to Say This Condition Has Nothing to Do with Accuracy#

Let me be clear about one thing. Using $H_u$ does not make the solution accurate. First-order upwinding still smears the interface. Even in the figure above, the step in $\alpha$ thickens with every step.

What the Abgrall criterion guarantees is a different kind of thing. When it is wrong, it is wrong in a physically sensible direction. A smeared interface is numerical diffusion, and that shrinks when you refine the grid. A 21 kPa spike standing in a uniform pressure field is not numerical diffusion. It does not shrink under refinement, it grows as the equation of state gets stiffer, and it often halts the computation with a negative pressure.

The same distinction came up in sharpening a smeared interface back, step after step. Some constraints can be bought off with accuracy, and some cannot. The Abgrall criterion is the latter. Move up to a high-order scheme and this condition still has to be satisfied separately.

Where I Look First the Next Time Pressure Spikes at an Interface#

If I meet this problem again, my order goes like this.

Run the uniform pressure and velocity problem first. If a problem with no waves shows a nonzero pressure variation, there is no need to look at the flux. The answer is in the nonconservative term or in the equation-of-state recovery.

Next, double the grid resolution and measure the same quantity. If the oscillation halves, it is a numerical diffusion problem. If it stays put, it is a consistency problem. This single run separates the two causes.

Finally, write out side by side which stencil each equation used for its nonconservative term. If mass used Rusanov, volume fraction used a central difference, and energy used something else again, that is the cause. The paper reusing a single $H_u$ across three equations was not a preference for shorter code.

References

• B. Re, R. Abgrall, Non-equilibrium Model for Weakly Compressible Multi-component Flows: the Hyperbolic Operator, arXiv:1911.00270 — §2.2 The discretization
• R. Abgrall, How to Prevent Pressure Oscillations in Multicomponent Flow Calculations: A Quasi Conservative Approach, J. Comput. Phys. 125 (1996)