Convection Schemes

In FVM, we need face values to compute convective fluxes: $F = \rho u \phi_f A_f$. But we store $\phi$ at cell centers. How do we get $\phi_f$? This interpolation choice — the convection scheme — profoundly affects accuracy and stability.

The Convection Problem

The Core Challenge

Consider the convective flux through the east face:

$$F_e = (\rho u)_e \phi_e A_e$$

We know $(\rho u)_e$ from continuity, and $A_e$ is geometric. But $\phi_e$ requires interpolation from neighboring cell values $\phi_P$ and $\phi_E$.

• Simple interpolation (central) → unstable for convection-dominated flows
• Stable schemes (upwind) → introduce numerical diffusion
• High-order schemes → accurate but may oscillate

The Peclet Number

The local Peclet number characterizes whether convection or diffusion dominates:

$$Pe = \frac{\rho u \Delta x}{\Gamma} = \frac{\text{Convection}}{\text{Diffusion}}$$

First-Order Upwind

The Idea

Use the upstream value for the face:

$$\phi_e = \begin{cases} \phi_P & \text{if } u_e > 0 \\ \phi_E & \text{if } u_e < 0 \end{cases}$$

Information travels in the flow direction — physically sensible!

Properties

• Unconditionally stable (bounded)
• Simple to implement
• Never produces oscillations

• Only first-order accurate $O(\Delta x)$
• Introduces numerical diffusion

Numerical Diffusion

Taylor series analysis shows upwind adds artificial diffusion:

$$\frac{\partial \phi}{\partial t} + u\frac{\partial \phi}{\partial x} = \underbrace{\frac{u \Delta x}{2}\frac{\partial^2 \phi}{\partial x^2}}_{\text{numerical diffusion}}$$

This smears sharp gradients, making fronts diffuse artificially.

$$\Gamma_{num} = \frac{\rho |u| \Delta x}{2}$$

On coarse meshes, this can exceed physical diffusion!

Central Difference

The Idea

Linear interpolation:

$$\phi_e = \frac{\phi_P + \phi_E}{2}$$

Properties

• Second-order accurate $O(\Delta x^2)$
• No numerical diffusion

• Unbounded — can oscillate
• Unstable for $Pe > 2$

Why Central Fails

For pure convection, central difference produces checker-boarding: alternating high-low values that the scheme cannot damp.

Higher-Order Schemes

QUICK (Quadratic Upstream Interpolation)

Uses three points for quadratic interpolation:

$$\phi_e = \frac{6}{8}\phi_P + \frac{3}{8}\phi_E - \frac{1}{8}\phi_W$$

(for $u > 0$, upstream-biased)

• Third-order accurate
• Less numerical diffusion than upwind
• Can still produce bounded oscillations

Second-Order Upwind

Uses two upstream points:

$$\phi_e = \frac{3}{2}\phi_P - \frac{1}{2}\phi_W$$

• Second-order accurate
• Less diffusion than first-order upwind
• Requires larger stencil

TVD Schemes (Total Variation Diminishing)

Blend between low and high-order schemes based on local gradients:

$$\phi_e = \phi_P + \frac{1}{2}\psi(r)(\phi_P - \phi_W)$$

Where $\psi(r)$ is a limiter function and $r$ is the gradient ratio.

• Minmod: $\psi = \max(0, \min(1, r))$
• Van Leer: $\psi = \frac{r + |r|}{1 + |r|}$
• Superbee: $\psi = \max(0, \min(2r, 1), \min(r, 2))$

• Preserve sharp gradients without oscillation
• Switch between first and second order
• Bounded by construction

Scheme Comparison

Deferred Correction

A practical approach combining stability and accuracy:

• Implicit: Use first-order upwind in the coefficient matrix
• Explicit: Add correction to source term: $S = S + (\phi_e^{HO} - \phi_e^{upwind})$

This maintains matrix diagonal dominance (stability) while approaching higher-order accuracy.

Practical Guidelines

Scheme Selection

Common Mistakes

• Using central for high-Pe flows: Oscillations, divergence
• Using upwind for mixing studies: Over-predicts mixing
• Ignoring numerical diffusion: Attributing it to physics
• Not checking convergence: Higher-order needs tighter tolerance

Numerical Diffusion Checklist

If results seem too diffusive:

• [ ] Is mesh fine enough in gradient regions?
• [ ] Is convection scheme order adequate?
• [ ] Are time steps too large (implicit smoothing)?
• [ ] Is turbulent diffusion physical or numerical?

Implementation in FVM

Coefficient Assembly

For face $e$ between cells P and E:

$$a_E = \max(-F_e, 0)$$

$$a_P += \max(F_e, 0)$$

$$a_E = D_e - F_e/2$$

$$a_P += D_e + F_e/2$$

Where:

• $F_e = (\rho u)_e A_e$ = face mass flux
• $D_e = \Gamma A_e / \delta_{PE}$ = diffusion coefficient

The Convection-Diffusion Equation

Combined convection and diffusion:

$$a_P \phi_P = a_E \phi_E + a_W \phi_W + S$$

Where coefficients depend on scheme choice.

Key Takeaways

• Convection schemes interpolate face values from cell-center data
• Peclet number determines whether flow is convection or diffusion dominated
• First-order upwind: Stable but introduces numerical diffusion
• Central difference: Accurate but unstable for Pe > 2
• Higher-order schemes (QUICK, TVD) balance accuracy and stability
• TVD limiters prevent oscillations while maintaining accuracy
• Numerical diffusion can mask physical phenomena on coarse meshes

What's Next

We've discretized convection and diffusion, but for incompressible flow, we still have the pressure-velocity coupling problem. The next lesson covers the SIMPLE algorithm and its variants — how to solve for pressure when there's no explicit pressure equation.