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Harmonic transmissibility across a one-dimensional material interface

Example

Let $Ω = (0, 1)$ with cells $K = (0, 1 / 2)$ and $L = (1 / 2, 1)$ , chosen centers $x_{K} = 1 / 4$ and $x_{L} = 3 / 4$ , cell coefficients $λ_{K} = 1$ and $λ_{L} = 9$ , and boundary values $u (0) = 0$ , $u (1) = 1$ . The stationary continuous solution has constant flux across the material interface. Compare the TPFA transmissibility using the two half-cell resistances with the transmissibility obtained by substituting the arithmetic coefficient mean.

Facts & Assumptions

Given: The two interval cells, piecewise-constant coefficient, Dirichlet boundary data, and the face transmissibility formula from Orthogonal TPFA diffusion with harmonic face transmissibility.

[F1]

The centers $x_{K} = 1 / 4$ and $x_{L} = 3 / 4$ are the choices specified in this example, permitted by Finite-volume cell-face meshes and geometric regularity; the shared interface is $x = 1 / 2$ .

[F2]

Orthogonal TPFA diffusion with harmonic face transmissibility: the interface transmissibility is $T_{F} = ∣ F ∣ / (d_{K, F} / λ_{K} + d_{L, F} / λ_{L})$ , with one shared flux $T_{F} (u_{K} - u_{L})$ outward from $K$ .

Verification

1.1givenF1algebra

The exact solution is piecewise affine. The constant $c$ in $λ u^{'} = c$ satisfies $1 = c (\frac{1}{2} / 1 + \frac{1}{2} / 9)$ , hence $c = 9 / 5$ . Thus $u (1 / 2) = 9 / 10$ , $u_{K} = u (1 / 4) = 9 / 20$ , and $u_{L} = u (3 / 4) = 19 / 20$ . The outward diffusive flux from $K$ is $- λ_{K} u_{K}^{'} = - 9 / 5$ and equals the outward flux from $L$ with the opposite orientation.

2.1F1F2step 1.1algebra

The half-distances are both $1 / 4$ , so [F2] gives $T_{F} = 1 / (1 / 4 + 1 / 36) = 18 / 5$ . The TPFA flux is $(18 / 5) (9 / 20 - 19 / 20) = - 9 / 5$ , exactly the continuous interface flux.

3.1F1step 1.1algebra∎

The arithmetic mean coefficient is $(1 + 9) / 2 = 5$ and the center distance is $1 / 2$ , which would give transmissibility $5 / (1 / 2) = 10$ . Its flux is $10 (9 / 20 - 19 / 20) = - 5$ , not $- 9 / 5$ . This finite interface calculation verifies that the resistance-weighted harmonic transmissibility is essential for this discontinuous-coefficient example.

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Sources

R. Eymard, T. Gallouët, and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, Vol. VII, 2000; complete updated author text (standard reference, not scraped)
Jérôme Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods, 2014 (standard reference, not scraped)