Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Boundary layer — where it appears

A thin region where the solution of a convection-dominated problem changes by an order-one amount, often thinner than one cell of the grid. Its width is set by the ratio of diffusion to convection, so refining the mesh eventually resolves it and no amount of algebra does.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

00.250.50.75100.51xuexact & tunedupwindcentralone of these is exacttuned, worst nodal error2.4·10⁻¹⁷upwind, worst nodal error0.14central, points outside [0, 1]16exact at every nodeand only at the nodes

The diffusion that makes the answer exact

Upwinding adds h/2 of artificial diffusion. Central differencing adds none. Add ε·ξ·Pe with ξ = coth(Pe) − 1/Pe and the computed solution is the exact one at every grid point, to 2.4·10⁻¹⁷ — at every Péclet number, on the problem it was derived from and on no other.

iterative · Convection 05101520253035404510⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²angle between the flow and the grid (degrees)worst nodal errortunedcentralupwindexact, and then nottuned, on the axis2.4·10⁻¹⁷tuned, five degrees off0.0079tuned at 45°0.066upwind at 45°0.0019fifteen orders of magnitude for five degreesand the worst of the three by forty-five

Exact along one axis

The tuned diffusion makes the answer exact at every node, and in two dimensions it holds at exactly one flow angle. Five degrees off the grid the relative error goes from 1.2·10⁻¹⁴ to 6.9, and by twenty degrees the scheme is worse than the upwinding it was built to improve on.

iterative · Convection worst nodal error, and nodes outside the interval the equation guarantees0°, streamline only2.39·10⁻¹⁷ · 0 outside0°, with crosswind5.25·10⁻¹⁷ · 0 outside15°, streamline only0.032 · 18 outside15°, with crosswind0.0145 · 9 outside30°, streamline only0.0571 · 50 outside30°, with crosswind0.0247 · 11 outside45°, streamline only0.0661 · 48 outside45°, with crosswind0.0281 · 0 outsidewhat the crosswind term buyserror ratio at 0°0.45error ratio at 15°2.2error ratio at 30°2.3error ratio at 45°2.3free where the scheme was exactand half the error everywhere else

The direction the diffusion does not go

Streamline diffusion adds τbbᵀ, a rank-one tensor that annihilates every direction across the flow. That is the design. The price is 18, 50 and 48 nodes where the computed solution leaves the interval the equation guarantees — and half a coefficient of crosswind diffusion halves the error at every angle while costing exactly nothing where the scheme was exact.

iterative · Convection 10²10⁻⁴10⁻³10⁻²10⁻¹grid points nworst nodal errorupwindtunedcentralthe same tuning, another problemtuned ÷ central at n = 3149tuned ÷ central at n = 12796central's error at the finest grid10·10⁻⁵exact on the problem it was derived fromand harmful on the one beside it

A parameter that is also a price

ξ = coth(Pe) − 1/Pe is the fraction of h/2 that makes a boundary-layer solution exact at every node. On a problem with no layer in it, the error the same scheme commits is ξ times upwinding's — 0.2511 against a ξ of 0.2504, 0.7461 against 0.7448 — so the number that buys the exactness is also the invoice.

iterative · Convection 00.7853981.57082.356193.1415900.250.50.751frequency θdamping |g(θ)|the oscillatory half →⅓ — the symmetric optimumno convectionwith convectionits imaginary parta modulus, not a valuesmoothing factor at this ω0.71best over every ω0.71the symmetric operator's, at ω = 2/30.33the imaginary part does not depend on ωso no ω removes it

A smoother that stops being one

Weighted Jacobi's smoothing factor on the convection–diffusion operator is a function of the cell Péclet number and nothing else — identical to eight digits at five grid sizes at matched Pe. It is 0.3335 at Pe = 0.016, exactly 1/√2 at Pe = 1, and 5.2190 at Pe = 7.8, where the sweep amplifies the modes it exists to remove.

iterative · Multigrid

Named alongside it

The objects these essays reach for when they reach for this one.

Convection diffusionPeclet numberArtificial diffusionStreamline diffusionM-matrixManufactured solutionUpwind differencingGrid alignmentUpwindingAnisotropyConvergence rateCrosswind diffusion

All concepts