State
Partial differential equation
A linear PDE on a continuous spatial domain, with the initial or boundary conditions that pin its solution — the problem as it is posed, before any grid exists. It is the one object here with infinitely many degrees of freedom, and that is what makes the first step real work rather than bookkeeping: replacing the continuum with finitely many numbers costs an error nobody can avoid, and that error is the first term in every budget downstream.
A state is an object you can be holding, named once so that two routes reaching the same thing are drawn as reaching the same thing. It says nothing about how you got here or where you can go next — that is entirely in the processes below.
This is a kind of
This state is not recorded as a kind of anything else. It stands on its own in the vocabulary.
Narrower kinds of this
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Records that are this object
Nothing in the catalogue has been joined to this state. That is a gap in the join rather than a claim that no such object exists; the shelf on /repository lists what is joined and what is not, with the reason.
Work that arrives here
No recorded process returns this. Either it is where a reader starts — a problem, a matrix, a machine — or it is an object this graph names and no route yet reaches.
Work that starts here
- Replace a spatial domain with a finite grid
Approximate the spatial derivatives of a PDE on finitely many points, leaving time continuous, so that what remains is a system of ordinary differential equations in the grid values. The method of lines: the continuum is gone, the clock is not.
- Graph-Laplacian finite differences
Discretize the domain onto a lattice and read the discrete Laplacian off the resulting graph: off-diagonal entries minus one between neighbours, each diagonal entry the degree of its vertex. Higher-order stencils are obtained by factorizing the operator through hypergraph incidence matrices, which is what lets the error fall faster than the second power of the spacing while keeping a form a simulator can consume.
- Discretize a PDE into one linear system
Replace every continuous variable at once — space together with time, or space together with velocity — so that the whole problem becomes a single matrix equation. Nothing is left to march: the grid values at every recorded point are unknowns of one system, solved in one go.