What are upwind schemes?

Methods that, when discretising convection, weight values from the upstream (upwind) side the flow comes from. First-order upwind never oscillates but has large numerical diffusion; second-order upwind and QUICK are more accurate but oscillate (overshoot) at sharp changes. TVD schemes sit between, suppressing oscillation with limiters.

Why does first-order upwind flatten peaks?

Its leading error term has the form of diffusion, equivalent to adding artificial viscosity. In the example, carrying a Gaussian once around on 100 cells left first-order upwind with 33% of the original peak, and a square wave's corners rounded with the maximum down to 0.68. Refinement reduces this error only in proportion to spacing.

How should a TVD scheme be chosen?

Use it to raise accuracy without oscillation. Minmod is safest but most diffusive (peak about 61%); van Leer is well balanced (75% peak, no overshoot); superbee keeps sharp changes crisp but tends to square off smooth peaks. In general-purpose codes, second-order upwind with a limiter is the standard choice.

Upwind Difference Scheme

Category: CFD Solver Methods | Revised 2026-10-01

CAE visualization for upwind schemes theory - technical simulation diagram

Upwind Differencing Scheme

Theory: how face values are formed

Overview

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My CFD settings offer first-order upwind, second-order upwind and QUICK for convection. Which should I pick?

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Finite volume needs face values to compute fluxes through cell faces; upwind schemes build them from upstream (upwind) cells. First-order upwind takes the upstream value as is — stable and non-oscillatory, but peaks flatten steadily. Second-order upwind and QUICK extrapolate from two or more upstream cells with lines or curves, more accurate but overshooting at sharp changes. The choice depends on what matters. Let's carry the same profile with each and compare.

Face-value formulas

$$ \phi_{i+1/2}^{UDS} = \phi_i,\quad \phi_{i+1/2}^{LUD} = \tfrac{3}{2}\phi_i - \tfrac{1}{2}\phi_{i-1},\quad \phi_{i+1/2}^{QUICK} = \tfrac{3}{8}\phi_{i+1} + \tfrac{3}{4}\phi_i - \tfrac{1}{8}\phi_{i-1} $$

For flow from left to right (side $i$ upstream). First-order upwind (UDS), linear upwind (LUD) and QUICK use progressively more points and higher accuracy.

Limited TVD schemes

$$ \phi_{i+1/2} = \phi_i + \tfrac{1}{2}\psi(r)\,(\phi_{i+1} - \phi_i),\quad r = \frac{\phi_i - \phi_{i-1}}{\phi_{i+1} - \phi_i},\quad \psi_{minmod} = \max(0, \min(1, r)),\ \ \psi_{vL} = \frac{r + |r|}{1 + |r|} $$

$r$ is the ratio of neighbouring gradients: near 1 where smooth, zero or negative near sharp changes or extrema, where the limiter $\psi$ weakens the second-order correction to prevent overshoot.

Coffee Break Trivia

Godunov's theorem

In 1959 the Soviet mathematician Godunov proved that non-oscillatory linear schemes can be at most first-order accurate — fixed-coefficient schemes can't combine high accuracy with freedom from oscillation. To overcome this barrier, 1970s–80s researchers devised nonlinear schemes whose coefficients adapt to the solution, notably van Leer's MUSCL and Harten's TVD. Limiters sidestep Godunov's theorem by switching between second order where smooth and first order where steep.

Worked example: carrying a peak and a square wave once around

On a periodic domain of 100 cells, with CFL 0.4 and two-stage Runge–Kutta, a square wave (height 1) and a Gaussian peak (height 1) were carried once around; ideally both return unchanged:

Scheme	Square max	Square min	Gaussian peak	Gaussian L1 error
First-order upwind	0.683	0.000	0.333	0.086
Second-order upwind	1.190	−0.215	0.746	0.064
QUICK	1.100	−0.101	0.926	0.009
TVD（minmod）	0.964	0.000	0.613	0.037
TVD（van Leer）	0.999	0.000	0.750	0.021

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QUICK preserves the peak best, yet overshoots the square wave by 10%.

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Right — accurate where smooth but oscillatory at steps. Concentrations or temperatures below zero or above bounds can break property calculations or drive reactions with negative concentration. Second-order upwind overshot the square wave by 19%. TVD keeps within 0–1 even at steps, and van Leer still keeps 75% of the peak. First-order upwind is safe but flattens the peak to a third. The basic split: QUICK or second-order upwind for accuracy in smooth flows, TVD for steps and sharp fronts.

Selection guidance

Start the first few hundred iterations with first-order upwind for stability, then switch to higher order.
Report final results at second order or higher (LUD, QUICK, TVD); don't stay first order.
Use TVD for bounded quantities: concentration, volume fraction, turbulence quantities.
Confirm grid independence with the scheme you actually use.
Grids aligned with the flow give less numerical diffusion.

Coffee Break Trivia

“Smoke that mixes too much”

A tunnel-ventilation CFD spread and diluted smoke far faster than in experiments. For stability, first-order upwind had been kept to the end, and its numerical diffusion added to the real turbulent diffusion. Switching to limited second-order upwind (TVD) kept the smoke layer distinct and brought concentrations close to experiment. Don't keep stabilising settings all the way to the reported results.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
First-order to the end	Mixing overestimated	Switch to second order+
QUICK for bounded quantities	Negative concentration or fraction	Use TVD
Mistaking wiggles for physics	Wrong conclusions	Check with another scheme
Over-restrictive limiter	Peaks clipped	Moderate limiters such as van Leer
Ignoring grid alignment	Diffusion in skewed flow	Flow-aligned grids

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I'd like to learn related topics.

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Try convection and diffusion with the convection–diffusion simulator. Related pages include the central difference scheme, finite volume fundamentals, time integration in CFD, density-based solvers and CFD mesh quality metrics.

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