A mass-drift certificate for spherical shallow-water patch coupling

Definitions and mathematical statement

Participant context. Item A.7 records Anne Gelb's interests in shallow-water equations for environmental fluid dynamics, global and local methods on spheres, and long-term simulations. It is not a posed problem: no PDE variant, discretization, assumptions or desired theorem are given. The aid develops a supplementary mass-balance diagnostic, which is not attributed as a theorem requested by the participant.

Reported space--time flux-register identity. Let a closed sphere have a nonoverlapping finite-volume partition into cells $K$ of areas $|K|$. Let $h_K^n$ be the cell-average water depth and \[ M^n=\sum_K|K|h_K^n. \] Assume no water-mass sources, no external boundary, and updates over a common synchronization interval of the form \[ |K|(h_K^{n+1}-h_K^n)=-\sum_{f\subseteq\partial K}\mathcal F_{K,f}^n, \] where $\mathcal F_{K,f}^n$ is the outward time-integrated mass-flux register, including face measure, all Runge--Kutta weights and all local substeps. Decompose nonconforming coarse/fine faces into common interface pieces and sum their registers over the same interval. For every paired interface $f=K|L$, put \[ \delta_f^n=\mathcal F_{K,f}^n+\mathcal F_{L,f}^n. \] The aid's exact algebraic identity is \[ M^{n+1}-M^n=-\sum_f\delta_f^n, \qquad M^N-M^0=-\sum_{n=0}^{N-1}\sum_f\delta_f^n. \] Matched interfaces have zero defect; the sum can therefore be restricted to global/local or coarse/fine interfaces at which registers are mismatched. In particular, \[ |M^N-M^0|\leq\sum_{n=0}^{N-1}\sum_f|\delta_f^n|. \] The triangle bound is sharp without cancellation information. Refluxing, meaning replacement/correction of a coarse register by the negative sum of the matching fine-side outward registers, makes each paired defect zero. It therefore gives exact mass conservation in algebraic arithmetic even under subcycling and Runge--Kutta stages, provided those same corrected integrated registers are used in the actual updates. Roundoff or other changes to the update are not silently excluded from a real implementation's measured error.

Reported hydrostatic interface calibration. At a face with bed values $b_L,b_R$, use the shared value $b_*=\max\{b_L,b_R\}$. For a possibly dry lake at rest with level $H$, define \[ h_i=(H-b_i)_+,\quad q_i=0,\quad \eta_i=h_i+b_i,\quad h_i^*=(\eta_i-b_*)_+\qquad(i=L,R). \] Then the two nonnegative reconstructed depths agree: \[ h_L^*=h_R^*=(H-b_*)_+. \] For a consistent numerical mass flux evaluated on these identical zero-velocity states, the face mass flux is zero, including wet/dry interfaces. Thus a stage at this reconstructed lake-at-rest state contributes zero mass register and zero interface defect. This statement alone is not a full momentum/source well-balancing proof: preservation of the complete lake state through every stage requires the rest of the scheme to preserve it as well.

This is a conservation and drift certificate for a specified class of updates. It does not certify accuracy, positivity of all evolved cell depths, momentum conservation, energy behavior or general long-time stability, and no numerical mesh or simulation is supplied.

Short English statement

Gelb's source item is research context. The aid's separate diagnostic identifies mass drift on a closed spherical finite-volume mesh exactly with accumulated interface-register mismatch. Shared refluxing cancels that drift, while a common max-bed reconstruction gives zero mass flux at a possibly dry lake-at-rest interface.

Sources

• Local UnsolvedMath registry, permanent ID $20002460$, catalogue entry, aim-pdes-notes.json, notes[71], workshop Stiff Sources and Numerical Methods for Conservation Laws, participant item A.7, Anne Gelb: https://aimath.org/WWN/balancelaws/balancelaws.pdf.
• Supplied research summary on finite-volume flux registers, synchronization drift, refluxing and shared max-bed reconstruction. No independent authoritative numerical-analysis citation or specific scheme is supplied.
• Attempt 1: context_only, not_a_problem, rigor "proved", low novelty confidence; the registry separately says open/OPEN-TRIAGE. These labels do not make the participant paragraph a posed question. The machine-generated aid is not peer reviewed and must be independently verified before citation; no source visit is claimed.