`vectorLaplacian`

definition

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module: IndisputableMonolith.NavierStokes.DiscreteNSOperator
domain: NavierStokes
line: 56 · github
papers citing: none yet

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IndisputableMonolith.NavierStokes.DiscreteNSOperator on GitHub at line 56.

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depends on

formal source

  53    (f : ScalarField siteCount) (x : Fin siteCount) : ℝ :=
  54  ∑ j : Axis, forwardDiff Λ h (backwardDiff Λ h f j) j x
  55
  56def vectorLaplacian {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)
  57    (u : VectorField siteCount) (x : Fin siteCount) (i : Axis) : ℝ :=
  58  scalarLaplacian Λ h (fun y => u y i) x
  59
  60def divergence {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)
  61    (u : VectorField siteCount) (x : Fin siteCount) : ℝ :=
  62  ∑ j : Axis, forwardDiff Λ h (fun y => u y j) j x
  63
  64def advection {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)
  65    (u : VectorField siteCount) (x : Fin siteCount) (i : Axis) : ℝ :=
  66  ∑ j : Axis, u x j * forwardDiff Λ h (fun y => u y i) j x
  67
  68def conservativeTransportField {siteCount : ℕ}
  69    (flux : ScalarField siteCount) (perm : Equiv.Perm (Fin siteCount)) :
  70    ScalarField siteCount :=
  71  fun i => flux i - flux (perm i)
  72
  73theorem total_conservativeTransportField_zero {siteCount : ℕ}
  74    (flux : ScalarField siteCount) (perm : Equiv.Perm (Fin siteCount)) :
  75    total (conservativeTransportField flux perm) = 0 := by
  76  unfold total conservativeTransportField
  77  rw [Finset.sum_sub_distrib]
  78  have hperm : (∑ i : Fin siteCount, flux (perm i)) = ∑ i : Fin siteCount, flux i := by
  79    simpa using (Equiv.sum_comp perm flux)
  80  rw [hperm]
  81  ring
  82
  83/-! ## Velocity Gradient Magnitude -/
  84
  85/-- Maximum absolute forward difference across all component/direction pairs. -/
  86def velocityGradientMag {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)

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