From 91334b5f69d9846bb3879207ec1252b4d8aa3f39 Mon Sep 17 00:00:00 2001 From: Emily Boudreaux Date: Sat, 19 Sep 2026 12:04:27 -0400 Subject: [PATCH] Clarify finite precision positivity of log density --- .../LOG_DENSITY_ASSESSMENT_2026-09-19.md | 21 ++++++++++++------- 1 file changed, 14 insertions(+), 7 deletions(-) diff --git a/experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md b/experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md index 3063fb0..3b46d91 100644 --- a/experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md +++ b/experiments/LOG_DENSITY_ASSESSMENT_2026-09-19.md @@ -17,8 +17,10 @@ eventual EOS to a barotrope. The important choice is what gets exponentiated: coordinate change on positive coefficient vectors. The measured negative density fields already have positive nodal coefficients. 2. Evaluating `rho = rho_scale * exp(u_h(x))` pointwise guarantees strictly - positive density wherever `u_h` is finite. This is a different approximation - family from polynomial density. Every density-dependent operator must use it. + positive density in exact arithmetic wherever `u_h` is finite. Floating-point + evaluation also needs checks for underflow and overflow. This is a different + approximation family from polynomial density. Every density-dependent + operator must use it. 3. A finite polynomial `u_h` cannot represent an exact zero-density vacuum surface with that raw exponential. A vanishing reference factor can encode the surface, but its asymptotic behavior becomes part of the representation. @@ -256,11 +258,16 @@ corresponding numbers are `6.65377`, `13.7153`, and `74.9786`. The first-order identity `delta rho/rho=delta u` makes a log step a relative density step. That can be useful across large density contrasts, but large log -steps create exponential changes. Underflow, overflow and weak response near -vacuum need explicit finite-value checks and meaningful globalization; silently -clamping the exponential would change the equations. A vanishing factor removes -the singular log profile when it matches the asymptotics, but retains the -density-weighted sensitivity near vacuum. +steps create exponential changes. Underflow can produce zero from finite log +coefficients, so finite-value checks alone do not ensure positivity. A raw +exponential implementation must also detect or reject underflow and nonpositive +reconstructed density in the material interior, including underflow when +multiplying by its reference scale or factor. Prescribed zero-density traces +from a vanishing reference factor remain intentional. Overflow and weak +response near vacuum also need explicit checks and meaningful globalization; +silently clamping the exponential would change the equations. A vanishing +factor removes the singular log profile when it matches the asymptotics, but +retains the density-weighted sensitivity near vacuum. With the explicit constant test retained, a converged weak exponential closure preserves its target cell mass under the closure quadrature. Exponentiation