335 lines
16 KiB
Markdown
335 lines
16 KiB
Markdown
# Independent log density: positivity, vacuum, and closure
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Branch: `research/log-density-assessment-20260919`.
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Parent: `7ed4d23`.
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Scope: mathematical review and a small uniform one-dimensional profile control.
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No stellar equilibrium, production operator, field order, normalization, or
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preconditioner was changed. This is separate from the density-only Q2/Q3/Q4 study.
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## Assessment
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Log density is compatible with **density remaining an independent physical
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unknown**. It need not make density a function of enthalpy or restrict the
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eventual EOS to a barotrope. The important choice is what gets exponentiated:
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1. Exponentiating the existing nodal coefficients and then evaluating the usual
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density polynomial does **not** guarantee a nonnegative field. It is only a
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coordinate change on positive coefficient vectors. The measured negative
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density fields already have positive nodal coefficients.
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2. Evaluating `rho = rho_scale * exp(u_h(x))` pointwise guarantees strictly
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positive density wherever `u_h` is finite. This is a different approximation
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family from polynomial density. Every density-dependent operator must use it.
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3. A finite polynomial `u_h` cannot represent an exact zero-density vacuum
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surface with that raw exponential. A vanishing reference factor can encode
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the surface, but its asymptotic behavior becomes part of the representation.
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The small control below shows that a raw exponential can satisfy all the chosen
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weak density moments while retaining an incorrect positive surface trace. Thus
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positivity and a small closure residual do not by themselves settle the vacuum
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or pointwise EOS question. Log density merits a separate formulation hypothesis;
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it is not a completed remedy for the current zero-pressure polytropic boundary.
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## Three distinct representations and their derivatives
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Let `q` denote geometry parameters, `h` enthalpy coefficients, and `z` the other
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unknowns. All variations below are at fixed reference coordinates; physical
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volume and other mapped geometry contributions must also be differentiated.
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### Logarithms of polynomial nodal coefficients
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Suppose the existing density field is
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\[
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\rho_h(\xi)=\sum_j r_j\varphi_j(\xi),\qquad
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r_j=\rho_*\exp(u_j),\quad \rho_*>0.
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\]
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For the existing residual vector `F(r,q,h,z)`, keeping its residual coordinates
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unchanged gives
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\[
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\widetilde F(u,q,h,z)=F(g(u),q,h,z),\qquad
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J_u=J_r\,\operatorname{diag}(r_j).
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\]
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This retains the original roots whose coefficient vectors are positive. The
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Gauss-Legendre nodal basis has sign-changing basis functions, so positive `r_j`
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does not imply nonnegative `rho_h`. In particular, taking logarithms of the
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already-positive coefficients at the measured n=1.5 root and applying this
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coordinate change would reproduce its negative polynomial exactly. Requiring
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positive coefficients alone does not remove that root.
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### A pointwise exponential of an independent FE field
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Instead choose
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\[
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u_h=\sum_j u_j\psi_j,\qquad
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\rho(\xi)=\rho_*\exp(u_h(\xi)).
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\]
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Now density is nonpolynomial, but remains independently parameterized by `u`.
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For a fixed positive scale,
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\[
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\delta\rho=\rho\,\delta u_h.
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\]
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Keeping the current density-moment tests as an explicit discretization choice
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would give, for the present barotropic closure,
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\[
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R_i=\int_{\Omega(q)}\varphi_i[\rho_*e^{u_h}-\rho_{\rm EOS}(h_h)]\,dV,
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\]
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\[
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\frac{\partial R_i}{\partial u_j}
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=\int\varphi_i\rho\psi_j\,dV,
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\qquad
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\frac{\partial R_i}{\partial h_k}
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=-\int\varphi_i\rho'_{\rm EOS}(h_h)\chi_k\,dV.
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\]
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With equal log trial and closure test bases, the density block is a
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density-weighted mass matrix. Positive quadrature weights and full column rank
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make it positive definite for strictly positive sampled density. Its magnitude
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can nevertheless become very small near vacuum.
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This is not the same residual obtained by minimizing the squared pointwise EOS
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error in the nonlinear trial family. Differentiating that least-squares
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objective introduces an additional `rho` in the test factor. That choice need
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not retain the constant density moment. The test space, residual and derivatives
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must therefore be chosen explicitly rather than inferred from the word “log.”
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For a general EOS, density can remain independent alongside temperature,
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internal energy and composition, as appropriate to the chosen physics. For
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example, an auxiliary pressure closure `P_aux - P(rho,T,X)` has log-density
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variation `-P_rho * rho * delta u_h`. No inversion to `rho(h)` is required by the
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log representation itself. The current barotropic Bernoulli equation is a
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separate modeling choice and does not automatically generalize to varying
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entropy or composition.
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### A vanishing reference factor
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Let
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\[
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\rho(\xi)=\rho_{\rm ref}(\xi,q)\exp(u_h(\xi)),\qquad
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\rho_{\rm ref}>0\text{ in the interior},\quad
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\rho_{\rm ref}=0\text{ on the intended vacuum surface}.
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\]
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Finite `u_h` now preserves both interior positivity and the prescribed zero
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trace. Its directional variation is
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\[
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\delta\rho=\rho\,\delta u_h+e^{u_h}\delta\rho_{\rm ref}.
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\]
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The second form is usable at a zero of the reference; division by
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`rho_ref` to form a logarithmic derivative is not. A reference fixed in mesh
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coordinates has no explicit shape derivative at fixed reference coordinates,
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but mapped volume, physical positions and force transformations still change.
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A reference defined from physical distance or updated with the accepted shape
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has additional dependencies that must be represented and differentiated.
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For a surface profile `rho ~ A s^n` and a reference `rho_ref ~ B s^m`,
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\[
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u\sim\log(A/B)+(n-m)\log s.
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\]
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The amplitude is bounded at the surface only when the leading exponent matches.
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A reference proportional to `s` is not a uniformly regular log coordinate for
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the n=1.5 or n=3 profiles. Choosing `rho_ref` from the current `rho_EOS(h)` would
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also introduce enthalpy dependence; it is not necessary for independent density
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and would need to be distinguished from a reference chosen independently of
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the live thermodynamic state. A fixed reference based on a model index is a
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reasonable polytropic experiment, not yet a general-EOS prescription.
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## Exact vacuum versus a finite-density outer boundary
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For the leading vacuum profile `rho=A s^n`, raw log density is
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`log(A/rho_scale)+n log(s)`. It tends to negative infinity, and its derivative
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is `n/s`. Moving density DOFs to interior Gauss points avoids evaluating the
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logarithm at zero, but does not give a finite polynomial the correct limit.
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Exponentiating that polynomial leaves a positive trace on the boundary.
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Imposing an additional exact `rho=0` condition on a finite raw log field is
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incompatible; it can lead to coefficients diverging rather than a finite root.
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The current weak closure can instead converge with a nonzero density trace,
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because matching volume moments does not enforce that trace. These are two
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different failure mechanisms, not a proof that the volume moment equations
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themselves are overdetermined.
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A physical outer boundary with specified positive density avoids this particular
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singularity. Such a boundary needs its own atmospheric/pressure prescription and
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EOS interpretation. Replacing a vacuum boundary by a numerical density floor
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would change the problem, not merely regularize coordinates. General EOS models
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can have different surface asymptotics, so the choice should follow the boundary
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physics rather than be hidden inside the linear solver.
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## Uniform surface-cell control
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`log_density_reference_model.py` compares independent approximations to `t^n`
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on `[0,1]`, for n=1,1.5,3. All densities below are normalized to the exact
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inner-edge value, which is one. Physical curvature, tangential variation,
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gravity, rotation and stellar equilibrium are absent.
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The raw log interpolant is the Q2 interpolant of `n log(t)` at the actual three
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Gauss-Legendre nodes. It is exponentiated at evaluation points, not interpolated
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again as density. The conservative exponential control solves only three small
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moment equations:
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\[
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\int_0^1\varphi_i(t)e^{u_2(t)}dt
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=\int_0^1\varphi_i(t)t^n dt,\qquad i=0,1,2.
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\]
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The tests are an L2-orthonormal polynomial basis; the right-hand moments are
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analytical. A three-coordinate Newton calculation fits these moments using a
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64-point rule and verifies them with an independently generated 128-point rule.
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This is a frozen toy fit, not a new stellar solve or a global existence proof.
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| n | Raw log interpolation: surface density | Raw log interpolation: relative L2 error | Moment-fitted exponential: surface density | Moment-fitted exponential: relative L2 error |
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|---|---:|---:|---:|---:|
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| 1 | 0.0615032 | 0.0669374 | 0.0673707 | 0.0578060 |
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| 1.5 | 0.0152527 | 0.0924952 | 0.0229264 | 0.0517531 |
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| 3 | 0.000232645 | 0.181788 | 0.00141091 | 0.0333959 |
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The moment fits take 3,4,5 tiny Newton updates, respectively; verified relative
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moment defects are at most `2.38e-14`. They are strictly positive everywhere in
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the cell, preserve mass to the reported arithmetic tolerance, and still have
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nonzero EOS error and an incorrect positive vacuum trace. This gives a concrete
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example in which positivity and weak closure convergence coexist with a boundary
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representation error.
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For comparison, the exact Q2 density projections have relative L2 errors
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`0`, `0.00952381`, and `0.05`, respectively. Thus positivity does not imply that
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the raw log family is a more accurate approximation at the same degree: the
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moment-fitted exponential improves this norm for the cubic profile, but worsens
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it for the n=1 and n=1.5 profiles. All these statements concern the one-dimensional
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profiles, not the actual stellar fields.
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A constant shift of the interpolated log field can enforce the exact cell mass:
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`u_new=u+log(M_target/M_current)`. It leaves the positive surface trace and
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nonconstant moment errors. The script records this distinct control; it is not
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used to alter a stellar state. A matched factor `t^n exp(0)` reproduces the target
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and its zero surface exactly by construction. That result demonstrates the role
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of the factor, not the general adequacy of an a priori reference profile.
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The smooth exponential controls have 64/128-point moment differences below
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`3e-16`. For the matched fractional factor `t^1.5`, the same quadratures differ
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by about `1.60e-10` in the largest orthonormal moment despite the represented
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function being exactly the target. That is integration error, not density
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approximation error. The script records it rather than replacing quadrature
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values by the known integral silently.
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## Conditioning and conservation
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For `rho=A Delta^n t^n`, using fixed reference polynomial bases on a cell of
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physical width `Delta` gives
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\[
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J_u=A\Delta^{n+1}H_n,\qquad
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(H_n)_{ij}=\int_0^1 t^n\varphi_i\varphi_j\,dt.
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\]
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The ordinary density mass matrix in this orthonormal reference basis is
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`Delta I`. Relative to it, the log-coordinate response carries the extra factor
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`A Delta^n`. Halving width therefore decreases the response scale by `2^-n`.
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This does not by itself worsen the condition number of the isolated cell matrix,
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but can increase scale disparities across cells and coupled blocks.
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The small weighted matrices for the exact reference profiles have condition
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numbers approximately `7.87298`, `16.8523`, and `94.5609` for n=1,1.5,3. These
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are three-dimensional local polynomial-space measurements, not estimates of the
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stellar Jacobian's condition number. For the fitted exponential controls the
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corresponding numbers are `6.65377`, `13.7153`, and `74.9786`.
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The first-order identity `delta rho/rho=delta u` makes a log step a relative
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density step. That can be useful across large density contrasts, but large log
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steps create exponential changes. Underflow, overflow and weak response near
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vacuum need explicit finite-value checks and meaningful globalization; silently
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clamping the exponential would change the equations. A vanishing factor removes
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the singular log profile when it matches the asymptotics, but retains the
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density-weighted sensitivity near vacuum.
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With the explicit constant test retained, a converged weak exponential closure
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preserves its target cell mass under the closure quadrature. Exponentiation
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alone does not ensure this; neither does positivity. Mass measured under a
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different rule, higher moments, inertia and virial balance still require their
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own consistency checks. A nonlinear change of trial family also does not
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automatically preserve a discrete variational identity coupling gravity,
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pressure and shape. Any integration must retain the true full residual and
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Jacobian tests and measure the physical consequences.
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## Architecture implications and recommended boundary of this work
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The natural internal abstraction is a typed independent-density representation
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with a physical-value evaluator and exact directional action. A log coordinate
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has no density units; its positive reference scale does. It should not be passed
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through APIs that assume its coefficients are physical density coefficients.
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The same representation must feed gravity source/force, centrifugal force,
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mass, angular momentum, diagnostics and exported profiles. Their dependency
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generation, quadrature, cache invalidation and normalization must agree.
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The current registry represents density as an independent scalar L2 field, and
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the prepared source and force operators read it as a polynomial grid function.
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The current field relationship catalog supports independent and differential
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relationships, not a nonlinear physical-value representation. Therefore a
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pointwise log field requires an explicit extension; setting a flag or
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exponentiating the accepted coefficient buffer is insufficient. Relevant local
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contracts are in `field_registry.cppm`, `field_base.cppm`,
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`prepared_gravity_source.cpp`, `prepared_barotropic_closure.cpp`,
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`normalization/stellar_equilibrium.cppm`, and
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`preconditioning/material_surface.cppm`.
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For the currently intended exact-vacuum polytrope, continue the separately
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authorized density-only order study first. Its cost and pointwise-error results
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provide the right baseline. If log density remains attractive, the next bounded
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question is whether a physically justified vanishing reference can provide a
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regular independent coordinate without importing a restrictive EOS assumption.
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For a genuinely finite-density outer boundary, raw pointwise log density is a
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different, potentially simpler candidate. Neither has been implemented or
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validated against a coupled star here; no logarithmic formulation, n=3
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convergence result, speedup, or virial improvement is claimed.
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## Primary-source context
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Vijaywargiya and Fu, *Two Finite Element Approaches for the Porous Medium
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Equation That Are Positivity Preserving and Energy Stable* (2024), formulate a
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pointwise exponential log-density method and explicitly discuss the infinite
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logarithm at zero density. Their compact-support implementation treats that
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issue through a degree-of-freedom deactivation strategy. Its time discretization,
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mass lumping and PDE-specific guarantees do not establish a corresponding
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method for steady stellar equilibrium. This is a relevant precedent and vacuum
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caution, not a proposed algorithm to transplant.
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[Primary paper](https://link.springer.com/article/10.1007/s10915-024-02642-x).
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Fu and Xu, *High-order space-time finite element methods for the
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Poisson-Nernst-Planck equations: Positivity and unconditional energy stability*,
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derive positive, conservative schemes through entropy variables and a specific
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space-time discretization. The conservation and stability results belong to
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that complete formulation, not to exponentiating coefficients in an arbitrary
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existing solver. [Author preprint](https://arxiv.org/abs/2105.01163).
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The representation identities, vacuum asymptotics and one-dimensional controls
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above are independent derivations rather than imported numerical evidence.
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## Reproduction
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From this checkout:
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```text
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/opt/homebrew/anaconda3/envs/general/bin/python experiments/log_density_reference_model.py --self-test
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/opt/homebrew/anaconda3/envs/general/bin/python experiments/log_density_reference_model.py --output ../../diagnostics/log_density_2026-09-19/reference_v1
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```
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The output directory must be fresh. It contains summary CSV/JSON and sampled
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profile CSV, with script and artifact SHA256 hashes. The retained invocation log
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is `../../diagnostics/log_density_reference_v1.log`. Four tiny test methods pass:
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Gauss moments, positive coefficient interpolation still admitting undershoot,
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moment-fit/vacuum behavior, and the weighted Jacobian directional derivative
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and small spectral identities. The production code was neither built nor run
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for this assessment.
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