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Uniform ideal-gas state is an unchanged numerical step

Physical scope

A symbolic one-step fixed-point calculation for a spatially uniform positive state in the specified periodic ideal-gas Euler model. It illustrates constant-state preservation only; it describes neither a shock nor a reported physical observation.

Example

For a dimensionless $γ > 1$ , define the admissible state set $G_{γ} = {(ρ, m, E) \in R^{3} : ρ > 0, E - \frac{m^{2}}{2 ρ} > 0} .$

Fix a dimensionless $γ > 1$ , $N \geq 2$ , $Δ x > 0$ metres, and $Δ t \geq 0$ seconds on a one-dimensional uniform periodic mesh. Choose one calorically perfect ideal-gas state $U = (ρ, m, E) \in G_{γ}$ , where $ρ > 0$ is mass density in kilograms per cubic metre, $m = ρ u$ is momentum density in kilograms per square metre second, $E = ρ (e + u^{2} / 2)$ is total energy density in joules per cubic metre, $u = m / ρ$ is velocity in metres per second, and $e > 0$ is specific internal energy in joules per kilogram. Set $p = (γ - 1) (E - m^{2} / (2 ρ)) > 0$ in pascals, $c = \sqrt{γ p / ρ}$ in metres per second, and $F (U) = (m, m^{2} / ρ + p, u (E + p))$ with mass, momentum, and energy flux components in kilograms per square metre second, pascals, and watts per square metre. Put $λ = Δ t / Δ x$ in seconds per metre. In every cell set $U_{i} = U$ , and at every face use the same speed $a > 0$ in metres per second with $a \geq ∣ u ∣ + c$ and $2 λ a \leq 1$ , the symmetric HLL flux ${\hat{F}}_{i + 1 / 2} = \frac{1}{2} (F (U_{i}) + F (U_{i + 1})) - \frac{a}{2} (U_{i + 1} - U_{i}),$ and the update $U_{i}^{+} = U_{i} - λ ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2}) .$ Every face flux equals $F (U)$ , so $U_{i}^{+} = U$ for every cell and the positive density and pressure remain unchanged. This uniform state has no shock; the calculation is a symbolic model example, not a reported measurement.

Example references

Positive first-order compressible model update — supplies the conditional one-step model interpretation and positivity hypotheses.

Facts & Assumptions

Given: The adopted ideal-gas model, one state $U \in G_{γ}$ , a periodic mesh with $N \geq 2$ , and the stated shared symmetric HLL flux and update.

[F1]

Positive first-order compressible model update: with $γ > 1$ , $N \geq 2$ , periodic states in $G_{γ}$ , symmetric face speeds $a \geq \max (∣ u_{L} ∣ + c_{L}, ∣ u_{R} ∣ + c_{R})$ , and $2 λ a \leq 1$ , the update uses $\hat{F} = \frac{1}{2} (F_{L} + F_{R}) - \frac{a}{2} (U_{R} - U_{L})$ and $U_{i}^{+} = U_{i} - λ ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2})$ .

Verification

1.1givenF1algebra

At every face, $U_{i} = U_{i + 1} = U$ , so substitution into [F1] gives ${\hat{F}}_{i + 1 / 2} = \frac{1}{2} (F (U) + F (U)) - \frac{a}{2} (U - U) = F (U) .$

2.1givenF1step 1.1algebra∎

Both periodic face fluxes adjacent to cell $i$ therefore equal $F (U)$ . Substitution into the update in [F1] gives $U_{i}^{+} = U - λ (F (U) - F (U)) = U$ . Since $U \in G_{γ}$ , its density and pressure are positive, and they are unchanged by the step. This is a symbolic model deduction.

Depends on

Positive first-order compressible model update