Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Manufactured solution — where it appears

A solution written down first, with the right-hand side computed from it, so that a scheme's error is known rather than estimated. It is the cheapest way to get an exact ground truth for a discretised problem, and it measures the discretisation rather than the arithmetic.

Named by 3 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 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

Named alongside it

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

Artificial diffusionBoundary layerConvection diffusionPeclet numberStreamline diffusionUpwind differencingAnisotropyConvergence rateDiscretisationDiscretisation errorGrid alignmentM-matrix

All concepts