Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Mass matrix — where it appears

The matrix of inner products of a discretisation's basis functions, which on a finite-difference grid is the diagonal of cell sizes. Projecting in the inner product it defines weighs each unknown by the mass it represents rather than counting nodes, and that is what keeps a discretised operator dissipative.

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

Péclet 100, orders 1, 2, 3, 6nodal edge at 1,000, outflow8.9weighted edge at 1,000-0.097unstable nodal models7110¹10²10³-20246810largest cell over smallestright edge of the numerical rangenodal, small cells at the outflownodal, small cells at the inflowcell-size weightedabove the dashed line the certificate is goneit is a sufficient condition, not a failure

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.

reduction · Reduced stability Péclet 100centred nodal edge at 1,0008.9upwind nodal edge at 1,0006.8upwind weighted edge at 1,000-0.45110¹10²10³-202468largest cell over smallestright edge of the numerical rangecentred, nodalupwind, nodalcentred, weightedupwind, weightedabove the dashed line the certificate is goneupwinding lowers the line and does not cross 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.

reduction · Reduced stability

Named alongside it

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

Convection diffusionGalerkin projectionNon-normalityNumerical rangeRational krylovReduced stabilityTransfer functionLyapunov equationPetrov–GalerkinUpwinding

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