Follow the direction of information on a grid

Try

For uₜ+c uₓ=0 with positive c, information arrives from the left. A spatial difference using the left neighbor respects that direction. With ν=cτ/h, the upwind update is Uⱼᵐ⁺¹=(1−ν)Uⱼᵐ+νUⱼ₋₁ᵐ.

The Courant number ν measures how many grid spacings a characteristic travels during one step. Its meaning is geometric before it is algebraic.

Predict

When ν=1, predict whether the update introduces interpolation error at grid points for constant-speed translation. When ν exceeds 1, can the stencil obtain information from far enough upstream in one step? Draw the backward characteristic and compare it with the two available grid locations.

Experiment

Take a center value 4 and left neighbor 0.

ν	Updated value	Interpretation
1/4	3	Convex interpolation
1	0	Exact one-cell shift
3/2	−2	Extrapolation and loss of monotonicity

At ν=1, repeated updates shift the sampled profile one cell each time, matching exact constant-speed advection at those times. For 0≤ν≤1, the nonnegative weights sum to 1, preserving local bounds. Negative speed requires the opposite spatial difference; the left-neighbor formula is not direction-independent.

Understand

The numerical domain of dependence must include the physical characteristic for a convergent approximation of arbitrary data. For this one-cell-per-step stencil, that gives |c|τ/h≤1 with the appropriate upwind direction. This CFL condition is necessary in the relevant limiting geometry, but not sufficient for every possible scheme.

Centered-space forward-time transport is a counterexample. Its amplification factor is G=1−iν sinθ, so |G|²=1+ν²sin²θ. For nonzero ν and generic θ, repeated steps amplify the mode. A small fixed Courant number does not make this explicit centered scheme stable over a fixed physical time as the grid is refined in the usual proportional way.

Positive-speed upwinding instead has G=1−ν+νe^(−iθ). For 0≤ν≤1, it is a convex combination of points on the unit circle and inside it, so |G|≤1. The scheme is first-order accurate for smooth solutions under ordinary coupled refinement. Its averaging introduces numerical diffusion, especially visible near sharp profiles; stable does not mean shape-preserving without error.

On a periodic grid, summing the update preserves the total sum because the shifted neighbor sum equals the original sum. More generally, conservative flux differences telescope to boundary fluxes. For nonlinear conservation laws, mass conservation alone does not select the entropy solution. Consistent monotone fluxes, stability or compactness and entropy arguments are needed for convergence theorems. The expansion-shock example from Burgers demonstrates why this extra selection matters.

Apply

Advect a short pulse with two admissible Courant numbers and compare its total mass, peak and width after the same physical time. Explain why conserved mass can coexist with a reduced peak. Then reverse the speed and identify the required stencil change before doing any arithmetic.

Continue

Coupled hyperbolic systems have several characteristic speeds. Symmetry supplies an energy estimate, while diagonalization reveals which combinations of fields travel in each direction.