For u = 2 m/s, dt = 0.04 s and dx = 0.1 m, what is the Courant number and what does it mean?

C = 0.8; the profile should move 0.8 of a cell per time step

C = 8; the profile should move 8 cells per time step

C = 0.08; the profile should move 0.08 of a cell per time step

C = 1.25; the profile should move 1.25 cells per time step

For flow in the +x direction, which expression gives the first-order upwind update?

phi_i - (C/2)(phi_(i+1) - phi_(i-1))

With the central (FTCS) scheme, node 1 becomes 292, below the old minimum of 300. What does this show?

An unphysical undershoot; forward Euler with central differencing is unconditionally unstable for convection

The scheme is bounded and the result is physically correct

The scheme is first order, so it adds numerical diffusion

What is the numerical diffusion coefficient of the upwind scheme here, and when does it vanish?

0.02 m^2/s; it vanishes when C = 1

0.1 m^2/s; it vanishes when C = 0

0.02 m^2/s; it vanishes when C = 0.5

0.2 m^2/s; it vanishes when C = 1

Practice · 6 marks

Upwind versus central scheme for convection

Short answer

Courant number C=uΔt/Δx=0.8. Upwind scheme (bounded, stable): φ₁=300, φ₂=304, φ₃=328, φ₄=368. Central difference (FTCS) gives unphysical undershoots: φ₁=292, φ₂=296, φ₃=328, φ₄=384. Upwind introduces numerical diffusion Γ_num=uΔx(1-C)/2=0.02 m²/s smearing gradients; central scheme is unconditionally unstable for pure convection.

For the linear convection equation $\partial\phi/\partial t+u\,\partial\phi/\partial x=0$ with $u = 2$ m/s, grid spacing $\Delta x = 0.1$ m and time step $\Delta t = 0.04$ s, find the Courant number. The values of $\phi$ at nodes 0 to 5 are 300, 300, 320, 360, 400, 400. Compute the new values at nodes 1 to 4 using (a) the first-order upwind scheme and (b) the central difference scheme in space with forward Euler in time. Comment on the results and on numerical diffusion.

Courant number

C=\frac{u\Delta t}{\Delta x}=\frac{2\times0.04}{0.1}=0.8

The physical shift in one step is $u\Delta t=0.08$ m, i.e. 0.8 of a cell, so the exact solution moves the profile to the right by 0.8 cell.

(a) First-order upwind (flow in +x, use the upstream node)

\phi_i^{n+1}=\phi_i^n-C(\phi_i^n-\phi_{i-1}^n)

$\phi_1=300-0.8(300-300)=300.0$
$\phi_2=320-0.8(320-300)=304.0$
$\phi_3=360-0.8(360-320)=328.0$
$\phi_4=400-0.8(400-360)=368.0$

(b) Central difference (FTCS)

\phi_i^{n+1}=\phi_i^n-\frac C2(\phi_{i+1}^n-\phi_{i-1}^n)

$\phi_1=300-0.4(320-300)=292.0$
$\phi_2=320-0.4(360-300)=296.0$
$\phi_3=360-0.4(400-320)=328.0$
$\phi_4=400-0.4(400-360)=384.0$

Node	1	2	3	4
Old	300	320	360	400
Upwind	300.0	304.0	328.0	368.0
Central	292.0	296.0	328.0	384.0

Comments

Upwind results stay between the old minimum (300) and maximum (400): the scheme is bounded and stable for $C\le1$ .
The central scheme gives 292 and 296 at nodes 1 and 2, below the lowest old value of 300 (an unphysical undershoot). Forward Euler with central space differencing is unconditionally unstable for pure convection, so these oscillations grow with time.
Upwind is first order. Its truncation error contains $\dfrac{u\Delta x}{2}(1-C)\dfrac{\partial^2\phi}{\partial x^2}$ , which acts like an extra diffusion (numerical or false diffusion) with coefficient

\Gamma_{num}=\frac{u\Delta x}{2}(1-C)=\frac{2\times0.1}{2}(1-0.8)=0.02\ \text{m}^2/\text{s}

that smears sharp gradients. For $C=1$ it vanishes and the scheme is exact in 1D.

Remedies: finer grids, higher-order bounded schemes (QUICK, TVD, second-order upwind) with limiters.

Answer: $C=0.8$ ; upwind 300, 304, 328, 368; central 292, 296, 328, 384; numerical diffusion 0.02 m²/s.

✓ Answer checked by Claude Opus 5.5 (high effort)

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Upwind versus central scheme for convection

Courant number

(a) First-order upwind (flow in +x, use the upstream node)

(b) Central difference (FTCS)

Comments

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