Concept

Grid alignment — where it appears

Whether a problem's difficult direction runs along a coordinate axis, which decides whether a repair tuned in one dimension survives at all.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as upwinding — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

Artificial diffusionBoundary layerConvection diffusionPeclet numberStreamline diffusionUpwindingAnisotropyCrosswind diffusionDiscrete maximum principleM matrixManufactured solutionNodal exactness

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