Convection diffusion — where it appears
Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.
A model that cannot be run
A stable system, reduced by matching its transfer function at three points exactly, comes back with a pole in the right half plane at four of nineteen placements — and matches at all three points to 1.3·10⁻¹⁴ while doing it. The construction did what it promised.
Half the conditions and a certificate
A two-sided interpolatory reduction of a stable convection–diffusion system came back unstable at four placements of nineteen. The one-sided reduction built from the same kind of solves came back stable at all 152 placements measured across eight Péclet numbers, and not by luck: every pole of a Galerkin model lies inside the numerical range of the operator, whose edge here is exactly the diffusion term's largest eigenvalue, −0.986 at Péclet 10. The price is the slope conditions, and spent as six one-sided shifts instead of three two-sided ones it beats the stable two-sided model at seven placements of ten.
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.
A certificate written in coordinates
Every one-sided reduced model of the convection–diffusion system was stable, because its poles cannot leave the numerical range and the range sat in the left half plane. Rescale the state by a diagonal matrix and nothing about the system changes — not its eigenvalues, not its transfer function, not one pole of any two-sided model — but the numerical range moves. At Péclet 40 a rescaling with condition number 260 pushes its edge to +6.6 and ten placements of nineteen return unstable one-sided models. Projecting in the inner product of a Lyapunov solution puts the certificate back in any coordinates.
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.
The inner product the mesh already computed
A Galerkin reduced model is certified stable when the operator's numerical range sits in the left half plane, and the certificate belongs to the coordinates. The coordinates that broke it before were a constructed rescaling. Grade a convection–diffusion mesh towards its outflow boundary layer — the grading anyone resolving the layer would choose — and nodal values break it too: the range's edge is past zero at a ratio of ten and reaches +8.9 at a thousand, and at Péclet 100 seven reduced models come back unstable. Weight the projection by the cell sizes the discretisation already computed and every model at every grading is stable, with the range's edge back at −0.098.
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.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Peclet numberArtificial diffusionBoundary layerM-matrixNon-normalityReduced stabilityTransfer functionUpwind differencingGalerkin projectionNumerical rangePetrov–GalerkinRational krylov