Give HLL's viscous energy flux a face velocity

[aliencaocao]

Fixes #2046.

Change

In src/simulation/m_riemann_solver_hll.fpp, the loop that exports the HLL interface velocity for the non-conservative terms also writes it to vel_src_rsx_vf when viscous = T. That array is the one the shared viscous routine reads for the τ·u work term. The velocity is the existing HLL trace. It is upwind outside the wave fan and averaged inside it:

Ψ_u = u_L                                   if 0 ≤ s_L
Ψ_u = (s_R u_L − s_L u_R) / (s_R − s_L)     if s_L ≤ 0 ≤ s_R
Ψ_u = u_R                                   if s_R ≤ 0

The formula holds for every velocity component. The condition that selected this loop is kept in one logical, need_nc_vel. The non-conservative export is therefore unchanged.

Why this is correct

MFC's energy equation carries the viscous work in its flux, ∇·[(ρE + p)u − τ·u] (equations.md, Section 2.1). With a consistent face velocity, the internal energy then gains exactly the dissipation Φ = τ:∇u ≥ 0 (Mei, MIT 1.63J lecture notes, Eqs. 4.1.16 and 4.1.18). Ψ_u is the HLL flux of the quantity whose "flux" is u. This is the interface velocity that MFC already uses for HLL's volume fraction and Kapila terms (HLL_HLLC_non_conservative_terms_derivations.md, Section 3). HLLC uses its own star velocity for the same work term. For symmetric wave speeds, it reduces to the arithmetic mean of u_L and u_R. That mean is the face velocity of a central viscous flux (Coralic and Colonius 2014, Section 3.2).

Before and after

The figure below shows the Gaussian vortex from the issue on 64², 128² and 256² cells. Each panel shows the heating rate over the first 0.005 time units against the analytic dissipation Φ = 4μA²(r/R)⁴ exp(−2r²/R²).

• Before: the core overheats, a ring at r ≈ 0.4 to 0.6 cools, and the relative RMS error is 112.9% to 113.9% on every grid.
• After: the heating ring of Φ appears, and no cell cools on any grid (minimum rate approximately −3 × 10⁻⁷, round off). The relative RMS error is 3.8%, 1.7%, 1.5% on 64², 128², 256².

Tests and goldens

The suite's Viscous cases all run the default HLLC solver. The one suite case that combines HLL with viscosity is the viscous_shock_tube example. The golden was regenerated on CPU with GCC 16.2 in double precision without case optimization:

• 70EC99CE 2D -> Example -> viscous_shock_tube

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Acknowledgement

• I confirm this PR meets the above expectations and reflects my own understanding and real-world context.

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PR template credit: junegunn

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[github-actions]

Claude Code Review

Head SHA: 7abeb33

Files changed:

• 1
• src/simulation/m_riemann_solver_hll.fpp

Findings:
No issues found. Checked in step 4: src/simulation/m_riemann_solver_hll.fpp (against m_riemann_state.fpp consumers of vel_src_rsx_vf and the HLLC writers).

[codecov]

Codecov Report

❌ Patch coverage is 28.57143% with 5 lines in your changes missing coverage. Please review.
✅ Project coverage is 61.89%. Comparing base (47568a0) to head (7abeb33).

Files with missing lines	Patch %	Lines
src/simulation/m_riemann_solver_hll.fpp	28.57%	3 Missing and 2 partials ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2054      +/-   ##
==========================================
- Coverage   61.90%   61.89%   -0.01%     
==========================================
  Files          86       86              
  Lines       22885    22888       +3     
  Branches     3369     3371       +2     
==========================================
+ Hits        14166    14167       +1     
- Misses       6220     6221       +1     
- Partials     2499     2500       +1     

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[github-actions]

Lines of Code

File	Lines	Diff
src/simulation/m_riemann_solver_hll.fpp	610	+3

Directory	Lines	Diff
simulation	28529	+3
total	47588	+3