5.3. Tracer Particles in DISPATCH — Design Context and Recommendations

5.3.1. 1. Problem Statement

Classical tracer particles advected with cell-centered velocity are biased because they do not represent numerical diffusion in finite-volume schemes.

This leads to:

incorrect mixing histories
systematic drift relative to fluid mass
unreliable “where did this mass come from?” diagnostics

We are evaluating improved methods for tracer transport and ancestry tracking.

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5.3.2. 2. Methods Under Consideration

5.3.2.1. 2.1 Classical tracers

ODE: dx/dt = u_cell
Pros: simple, cheap
Cons: ignores numerical diffusion → biased trajectories

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5.3.2.2. 2.2 Monte Carlo tracers (AREPO-style)

Particles jump between cells with probabilities proportional to mass flux
Pros: unbiased transport
Cons: discontinuous trajectories, noisy

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5.3.2.3. 2.3 Itô tracers (recommended baseline)

Continuous stochastic differential equation:

dx = u_ito * dt + sqrt(2 * kappa_ito * dt) * random_kick

Where:

drift u_ito is derived from mass fluxes
diffusion kappa_ito matches numerical diffusion

Pros:

continuous trajectories
statistically correct
local (patch-friendly)
relatively simple to implement

Cons:

stochastic noise

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5.3.2.4. 2.4 Flux-consistent (PIC-style / Esirkepov-like)

Enforces:

particle flux = finite-volume mass flux

Pros:

exact conservation
deterministic

Cons:

complex bookkeeping
difficult with AMR + async stepping
unnecessary for passive tracers

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5.3.3. 3. Key Conclusion on Conservation

Exact particle-level conservation is not required.

Reason:

tracers are diagnostic only
finite-volume solver already conserves mass
resampling (merge/split) breaks exact conservation anyway

What matters is:

E[rho_particles] = rho_grid

i.e. statistical correctness, not exact matching

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5.3.4. 4. Core Challenge: Ancestry Tracking

Goal:

“Where did the mass in this cell come from?”

Complication:

particles split and merge
one particle may represent mass originating from many regions
ancestry becomes a many-to-many mapping

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5.3.5. 5. Correct Conceptual Model

A tracer particle is:

particle = (position x, weight w, ancestry A)

Where:

A is a distribution over origins

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5.3.6. 6. Ancestry Handling Under Split/Merge

Split:

parent: (w, A)

child1: (alpha * w, A)
child2: ((1 - alpha) * w, A)

Merge:

w_new = w1 + w2
A_new = (w1 * A1 + w2 * A2) / w_new

This is equivalent to mixing passive scalars.

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5.3.7. 7. Practical Representations of Ancestry

5.3.7.1. Option A — Full genealogy (exact, expensive)

Store full directed acyclic graph:

child -> parents + weights

Not scalable.

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5.3.7.2. Option B — Sparse origin vectors

Each particle stores:

integer :: origin_id(kmax)
real    :: fraction(kmax)

Merge:

weighted union
truncate to largest contributors

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5.3.7.3. Option C — Sample-based ancestry (recommended)

Each particle stores:

integer :: origin_sample(K)

Merge:

resample from parents proportional to weight

Pros:

bounded memory
unbiased estimate
robust under resampling

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5.3.8. 8. Snapshot-Based History (Recommended Architecture)

Do not store full history continuously.

At checkpoints, store:

particle_id
position
weight
ancestry samples

Between checkpoints (optional), store events:

split / merge / transfer

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5.3.9. 9. Backward Query Algorithm

To answer:

“Where did mass in cell C at time T come from?”

Procedure:

collect particles contributing to cell C at time T
read ancestry samples
optionally recurse via event logs to earlier checkpoints

This makes history reconstruction an offline operation.

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5.3.10. 10. Recommended DISPATCH Implementation

5.3.10.1. Phase 1 (baseline ~1 week)

implement Itô-2 tracer update
add particle weights
add ancestry samples (K ~ 4–16)

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5.3.10.2. Phase 2 (~2–4 weeks)

implement merge/split resampling
implement checkpoint snapshots
implement ancestry query tools

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5.3.10.3. Phase 3 (optional)

SGS-Itô (turbulent diffusion)
selective genealogy logging
variance reduction strategies

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5.3.11. 11. What NOT to do initially

no global residual correction (non-local, expensive)
no strict flux-conserving particle transport
no full genealogy for all particles

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5.3.12. 12. Summary

classical tracers are biased
Itô tracers correct transport statistically
exact conservation is unnecessary
ancestry must be treated as a distribution

Best approach:

weighted particles
stochastic transport
sampled ancestry
snapshot-based history

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5.3.13. 13. Minimal Codex Task Specification

Implement tracer mode ito2 in DISPATCH:

compute drift and diffusion from face mass fluxes
update particles using Euler–Maruyama
store particle weights
maintain K ancestry samples per particle
implement merge/split via resampling
output snapshots including ancestry samples