VelCoeff

formal source

  53-- in `EuclideanSpace`, not plain Pi types (which carry the sup norm).
  54
  55/-- Velocity Fourier coefficient at a mode: a Euclidean real 2-vector (û₁, û₂). -/
  56abbrev VelCoeff : Type := EuclideanSpace ℝ (Fin 2)
  57
  58/-- The Galerkin state: velocity coefficients for each truncated mode and each component. -/
  59abbrev GalerkinState (N : ℕ) : Type :=
  60  EuclideanSpace ℝ ((modes N) × Fin 2)
  61
  62/-!
  63## Discrete dynamics
  64-/
  65
  66/-- Squared wave number \( |k|^2 \) as a real number. -/
  67noncomputable def kSq (k : Mode2) : ℝ :=
  68  (k.1 : ℝ) ^ 2 + (k.2 : ℝ) ^ 2
  69
  70/-- Fourier Laplacian on coefficients: (Δ û)(k) = -|k|² û(k). -/
  71noncomputable def laplacianCoeff {N : ℕ} (u : GalerkinState N) : GalerkinState N :=
  72  WithLp.toLp 2 (fun kc => (-kSq ((kc.1 : Mode2))) * u kc)
  73
  74/-- Abstract Galerkin convection operator (projected nonlinearity).
  75
  76Later we will replace this with the explicit Fourier convolution + Leray projection. -/
  77def ConvectionOp (N : ℕ) : Type :=
  78  GalerkinState N → GalerkinState N → GalerkinState N
  79
  80/-- Discrete Navier–Stokes RHS: u' = νΔu - B(u,u). -/
  81noncomputable def galerkinNSRHS {N : ℕ} (ν : ℝ) (B : ConvectionOp N) (u : GalerkinState N) :
  82    GalerkinState N :=
  83  (ν • laplacianCoeff u) - (B u u)
  84
  85/-!
  86## Energy functional and inviscid conservation