A weight is not a location

The edge sits where the cells meet the flow answered a question about where. A convection–diffusion operator discretised on a mesh whose cells shrink towards the outflow, where a boundary layer forms, has a numerical range in nodal coordinates that crosses into the right half plane as soon as the grading reaches ten to one. The direction that attains the range’s edge — the top eigenvector of the operator’s symmetric part — was expected in the layer, among the smallest cells. It was not. It peaked where the cell Péclet number $h_i\,\mathrm{Pe}/2$ was between 0.46 and 1.24: in the transition, where diffusion and convection are comparable and a centred stencil’s convection part is no longer small beside its diffusion part.

The inner product the mesh already computed had already found the repair. Weight the inner product by each node’s cell size — the diagonal mass matrix the discretisation forms anyway — and the range is dissipative at every grading, Péclet number and direction measured, and every Galerkin model built in that inner product is stable. Put together, the two findings suggested an economy, and the essay that found the edge’s location signed it: “Weight the inner product by cell size over the nodes where the cell Péclet number is between a quarter and two, and by one elsewhere. The prediction with a sign is that on every grading at Péclet 40 and 100 this local weight makes the range dissipative, and that the Galerkin models built in it fail on none of the placements where the nodal ones failed — so that the full mass weighting’s success comes entirely from the transition the eigenvector occupies.”

Both halves fail, badly, and the reason is that a weight does its work by being the right weight everywhere, not by being present where the trouble is.

Four inner products on one operator #

The operator is the earlier essays’: thirty interior nodes on $(0, 1)$, cells growing geometrically so that the largest is 1 to 1,000 times the smallest, small at the outflow; centred differences for both terms, so that on a uniform mesh the convection matrix is exactly skew. Each inner product is a diagonal weight $p_i$. Nodal is one everywhere. Cell sizes everywhere is $(n + 1)\,m_i$, the mass matrix scaled so that a uniform mesh’s weight is one. Band only is the prediction’s: the scaled cell size where the cell Péclet number is between a quarter and two, one elsewhere. And cell sizes outside the band is its reverse, which was added to tell a weight that matters by location from one that matters by shape.

On a uniform mesh all four weights are one and give the same edge, $-0.10$ at Péclet 100. The cell sizes everywhere then hold it there: $-0.097$ to $-0.099$ at every grading. The nodal edge climbs to 1.4 at a grading of ten and 8.9 at a thousand, which is the failure the two earlier essays measured. The band-only edge is 0.78 at ten — better than nodal, for once — and then jumps: 114 at thirty, 153 at a hundred, 181 at three hundred, 144 at a thousand. At Péclet 40 the same pattern is a little milder, 6.2 at ten and 9.1 to 14 from thirty on, against nodal edges of 0.9 to 4.2. At every grading from ten, at both Péclet numbers, the band-only weight puts the range further right than having no weight at all, and from thirty it does so by a factor of three to forty.

The edge moves to the step #

The edge eigenvector says where the new trouble is.

In nodal coordinates the edge eigenvector is a broad hump centred on node 16, inside the band, where the earlier essay put it. In the band-only inner product it is a sharp spike at nodes 22 and 23, just downstream of the band, with 91 per cent of its weight within a node or two of the place where the weight steps from 0.16 back up to one. The band begins well upstream, at node 9, where the weight steps from one up to 1.2; the downstream step is the larger by far, because the cells inside the band have already shrunk by a factor of seven by its far end.

At Péclet 100 the step holds 89 to 94 per cent of the edge eigenvector at every grading from thirty; at Péclet 40, where the band is wider and its downstream step smaller, 65 to 71 per cent. The prediction located the old edge correctly and moved it, but the weight that moved it put a new edge of its own at its own boundary, and that new edge is fifteen to forty times further right.

The one diagonal that makes convection skew #

Why a step in the weight produces an edge is a matter of two lines of algebra, and the algebra says more than the measurement could. The centred convection stencil at node $i$ is $(u_{i+1} - u_{i-1})/s_i$ with $s_i = h_i + h_{i+1}$ the two cells beside the node. In the inner product weighted by $p$, the symmetric part of the convection operator has, between nodes $i$ and $i+1$, the entry

$$\tfrac12\Big(\frac{p_{i+1}}{s_{i+1}} - \frac{p_i}{s_i}\Big),$$

and it vanishes for every pair exactly when $p_i/s_i$ is the same at every node — when $p$ is proportional to the cell sizes. So the mass weight is not one good weight among many; it is the only diagonal weight in which the convection term is skew and contributes nothing to the range’s edge, and in every other diagonal weight the convection term has a symmetric part proportional to the weight’s departure from it. That symmetric part is multiplied by the convection’s coefficient, one, while the diffusion that would have to absorb it is multiplied by $1/\mathrm{Pe}$.

On a nodal weight the departure is spread over the whole mesh, a steady ratio of neighbouring cell sizes. On the band-only weight it is zero inside the band and outside it, and concentrated at the two steps, where $p_i/s_i$ changes by a factor of six in one node. That is a larger local departure than the nodal weight ever makes, and the edge finds it.

Both terms need the cell sizes #

The algebra above is about convection, and the diffusion term has the same structure on a graded mesh: its non-uniform second difference is symmetric only after every row is multiplied by the node’s cell size, and only then is it negative definite. So the cell sizes are the one diagonal weight for both terms at once, and splitting the operator shows what each weight does to each.

Taken alone, the convection term’s range edge is zero to rounding in the cell-size weight — $3 \cdot 10^{-14}$ at a grading of a hundred — and 33 in nodal coordinates and 117 in the band-only weight. The diffusion term’s edge is negative in both the cell-size and the nodal weights, $-0.098$ and $-0.076$: in nodal coordinates the second difference is not symmetric, but its symmetric part stays negative, and it is the diffusion that pulls the nodal operator’s edge from 33 back to 4.7. In the band-only weight the diffusion term’s own edge is $+72$ at a grading of a hundred and $+63$ at a thousand. The step that breaks the convection term’s skewness breaks the diffusion term’s definiteness too, at the same node, and the two add instead of cancelling: 117 and 72 make the 153 the full operator shows. That is why the band-only weight does so much worse than no weight at all. Nodal coordinates are wrong for convection by a steady ratio everywhere and right enough for diffusion; the band-only weight is wrong for both, by a factor of seven, in one place.

The eigenvalues that are not there is the general warning behind all of this: the range and the pseudospectra of a non-normal operator move with any change of coordinates while its eigenvalues do not, and a change that looks small in one row can move them a long way. Here the change is a factor of seven in one entry of a diagonal, and it moves the range’s edge by a hundred and fifty.

Ramping the steps #

If the trouble is the step, a ramp should soften it: let the weight’s exponent on the cell size fall from one at the band’s edge to zero over $r$ nodes outside it, so that the weight passes from cell sizes to one gradually.

It softens it slowly. At Péclet 100 and a grading of a hundred the edge falls from 153 with no ramp to 81, 50 and 11 with ramps of two, four and eight nodes, and to 0.45 with a ramp of sixteen — still positive, and on a mesh of thirty nodes a ramp of sixteen either side of a band fourteen nodes wide is the cell-size weight over nearly the whole mesh. At a grading of a thousand the sixteen-node ramp leaves 2.9 at Péclet 100 and 5.1 at Péclet 40. Only at a grading of ten, at Péclet 40, does a ramp of eight nodes bring the edge below zero. The algebra predicted this: a ramp spreads the departure from the cell sizes over more nodes, it does not remove it, and every node where $p_i/s_i$ is not constant contributes.

A certificate is not the stability #

The band-only weight’s models do fail, as its range said they might.

At Péclet 100 they fail on 2, 6 and 5 placements at gradings of 100, 300 and 1000; the nodal models fail on 4, 5, 7 and 7 from thirty; the models weighted by cell sizes everywhere on none. At Péclet 40, one band-only model fails at a hundred and one at a thousand, against two and four nodal ones. The band-only weight is better than nothing at failing — thirteen failures at Péclet 100 to the nodal twenty-three — despite a range edge fifteen to forty times further right. And the reverse weight, cell sizes everywhere but the band, has a positive edge at every grading from three and not one failed model at either Péclet number.

The failures are nearly all at one order. Of the band-only weight’s thirteen at Péclet 100, eleven are second-order models and two are sixth-order; the nodal weight’s twenty-three split twenty-one and two in the same way. No first- or third-order model fails in any weight. A second-order model has a two-dimensional subspace and two poles, and the projection puts one of them near the range’s edge whenever the subspace leans towards the edge direction; a third-order one, built with a third interpolation point, evidently leans away. The pattern is the same in both weights, which says the band-only weight changed how far the edge is and not which models are exposed to it.

That gap between the certificate and the models is the earlier finding of half the conditions and a certificate seen from its other side. A range in the left half plane guarantees that every Galerkin model is stable; a range in the right half plane guarantees nothing either way, and the models fail only where a projection subspace happens to align with the direction that attains the edge. The band-only weight’s edge direction is a spike at one step, which the Krylov subspaces the interpolation builds reach less often than the nodal edge’s broad hump in the transition. So the band-only weight fails less often than nodal and is still not a certificate, and the reverse weight never fails and is not one either. What the prediction asked for — dissipative everywhere, failing nowhere — only the cell sizes deliver, and they deliver both.

What the location was worth #

The earlier essay’s location was right. The nodal edge does live in the transition, and its eigenvector is a hump over the nodes the band selects. What the prediction inferred from it — that a weight could be applied where the edge lives and nowhere else — treats the weight as a repair to a region. The algebra treats it as a change of coordinates for the whole operator, and the convection term is skew in exactly one set of diagonal coordinates. A certificate written in coordinates found that a stability certificate belongs to the coordinates it is stated in rather than to the operator; this is the sharper version of that for this operator: the certificate has exactly one diagonal coordinate system, and any departure from it, anywhere, writes a new edge where the departure is.

The smaller cell downstream found that upwinding does not rescue the nodal range either, because every row of the upwind stencil’s symmetric part carries the difference of neighbouring cell widths. That was the same algebra in a different stencil: the stencil is skew only in the coordinates its own cell widths define. A model that cannot be run, the first essay on these reduced models, found a two-sided interpolatory model unstable at four placements of nineteen on a uniform mesh, with nothing graded at all, and showed that the mesh is not the only route to the right half plane; but it is the route these weights were asked to close.

The prediction is in good company in its mistake. The diffusion that makes the answer exact found an artificial diffusion that makes a one-dimensional convection–diffusion solution exact at every node, and it works because it is the right amount at every node, not because it is applied where the solution is steep. Applied only in the layer, it would be exact nowhere. The cell-size weight belongs to the same family of exact corrections: it is right because it is computed from the stencil node by node, and the place where the error would otherwise show is not the place where the correction does its work.

What thirty nodes do not show #

Thirty nodes make a band of nine to thirty of them, and steps that are one node wide. On a fine mesh the same band would span more nodes and its steps would be the same factor in a node of smaller width, so the departure per node would be the same and the symmetric part at the step would scale with the stencil’s $1/h$ — larger, not smaller. That is an argument, not a measurement. The scheme is centred; an upwind scheme’s convection part is not skew in any diagonal weight, as the earlier essay found, and the question of a local weight does not arise there in the same form. And the band’s limits, a quarter and two, are the prediction’s; a band from a half to one, the earlier essay’s measured range for the edge’s peak, would be narrower, its steps the same size, and its edge no better.

Still open: a weight from the stencil, and a finer mesh #

A weight built from the stencil rather than the mesh. If the only skew-making diagonal weight is the one proportional to $s_i$, then on a mesh whose grading is not geometric — a mesh adapted to a solution — the cell sizes are still that weight, while a weight read off the solution’s gradient is not. The prediction with a sign is that on a mesh refined by equidistributing the boundary layer’s arc length, the mass weight’s edge stays at $-1/\mathrm{Pe}$ to within ten per cent at Péclet 40 and 100, and a weight proportional to the local cell Péclet number, which matches it only where the mesh is geometric, has a positive edge at every refinement level.

Refinement. On 120 nodes with the same grading the band covers four times as many nodes. The prediction is that the nodal edge eigenvector still peaks where the cell Péclet number is between a half and one, that the band-only weight’s edge grows in proportion to the inverse width of the cells at its downstream step, and that the edge with cell sizes everywhere stays at $-1/\mathrm{Pe}$ — so that refinement makes every weight but one worse, at the places the algebra names.