Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Upwind differencing — where it appears

Differencing the transport term against the flow, which never oscillates and is solving a problem with more diffusion than the one posed. It is exactly central differencing applied to a problem with more diffusion in it, so the stability is bought by solving a different equation.

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

00.250.50.75100.250.50.751xuexactcentral differencesupwindthe oscillation is exact‖Ax − b‖/‖b‖ for the central answer4.6·10⁻¹⁸values outside [0, 1]16worst excursion0.52the dashed lines are 0 and 1, which the equation guaranteesno solver was involved

The stencil that is not symmetric

Past a cell Péclet number of exactly one — measured by bisection at 1.0000000000000002 — the central-difference solution of a convection–diffusion problem oscillates from point to point and leaves the interval the equation guarantees, at 16 of 31 grid points. It is the exact solution of its own linear system, to 4.6·10⁻¹⁸. No solver was involved.

iterative · Convection 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 00.250.50.75100.250.50.751xuexact at ε = 0.005exact at ε(1 + Pe)the upwind answeran exact answer to a different question‖upwind(ε) − central(ε(1+Pe))‖/‖·‖0distance to the problem it solves0.026distance to the problem posed0.36the added diffusion is h/2 = 0.001953, whatever ε isso refining removes it

A different equation on every grid

Upwinding is the exact discretisation of a convection–diffusion problem with diffusion ε + h/2, entry for entry, at a relative difference of between 0 and 1.26·10⁻¹⁶ on every mesh from 15 points to 511. The equation it is exact for is chosen by the mesh and not by ε — the added diffusion is 0.01563 on a 31-point grid whether ε is 0.2 or 0.001.

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 diffusionM-matrixBoundary layerDiscretisationManufactured solutionSmoothing factorStreamline diffusionBackward errorConvergence rateDiscretisation error

All concepts