Upwinding — where it appears
Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.
The smaller cell downstream
Nodal values on a convection–diffusion mesh graded towards its outflow layer put the numerical range in the right half plane, and Galerkin reduced models then fail. A code that grades towards the layer usually also upwinds, and upwinding adds diffusion. It does not give the certificate back: at Péclet 100 and a grading of a thousand the range's edge falls from 8.94 to 6.78 and stays positive, and 18 reduced models fail where 23 did. Each row of the upwind stencil's symmetric part sums to half the difference between the inverse widths of the cell after the node and the cell before it, so wherever the cells shrink downstream the dissipation upwinding adds is outweighed by the coupling to the smaller cell next door.
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.
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.
The edge sits where the cells meet the flow
On a convection–diffusion mesh graded towards its outflow, the nodal numerical range reaches into the right half plane, and Galerkin reduced models fail. The account was that the edge lives in the cells where diffusion and convection are comparable, and drawing the eigenvector that attains it says the account is right: on every operator whose mesh has a cell Péclet number above one, the eigenvector peaks where that number is between a half and one and a quarter, just downstream of where it crosses one, and the outflow layer's five smallest cells hold under four per cent of it. The other question has a short answer too. A one-sided stencil scaled by a cell width is dissipative in nodal coordinates only if its weights do not grow downstream, and a consistent one must grow there: the only dissipative weight is the constant one, which is wrong by up to four units of the derivative.
Named alongside it
The objects these essays reach for when they reach for this one.
Convection diffusionPeclet numberArtificial diffusionBoundary layerGalerkin projectionGrid alignmentMass matrixNon-normalityNumerical rangeReduced stabilityStreamline diffusionAnisotropy