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163 lines
6.9 KiB
Text
163 lines
6.9 KiB
Text
/-
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KdVBurgersPDE.lean — Compound KdV-Burgers Equation in Q16_16
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u_t + α·u·u_x + β·u_xx + γ·u_xxx = 0
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Combines nonlinear advection (α), viscous diffusion (β), and
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third-order dispersion (γ). Captures both shock formation
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and wave-breaking phenomena.
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Reference:
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- Wang 1996 (10.1016/0375-9601(96)00103-x) — Exact solutions for compound KdV-Burgers
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- Feng 2002 (10.1088/0305-4470/35/2/312) — First-integral method
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-/
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import Semantics.FixedPoint
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import Semantics.BurgersPDE
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namespace Semantics.KdVBurgersPDE
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open Semantics.FixedPoint
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open Semantics.FixedPoint.Q16_16
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-- ============================================================
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-- 1. KdV-BURGERS STATE
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-- ============================================================
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/-- Discrete KdV-Burgers state with dispersion coefficient γ -/
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structure KdVBurgersState where
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base : Semantics.BurgersPDE.BurgersState -- N, u, ν, dx, dt, t
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α : Q16_16 -- nonlinear coefficient (advection strength)
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β : Q16_16 -- viscous coefficient (diffusion)
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γ : Q16_16 -- dispersive coefficient (KdV term)
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deriving Repr, Inhabited
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-- ============================================================
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-- 2. THIRD-ORDER DIFFERENCE OPERATOR
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-- ============================================================
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/-- Third derivative approximation: (u[i+2] - 2u[i+1] + 2u[i-1] - u[i-2]) / (2·dx³) -/
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def thirdDiff (u : Array Q16_16) (i : Nat) (dx : Q16_16) : Q16_16 :=
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if h1 : i > 1 then
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if h2 : i + 2 < u.size then
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let uim2 := u[i-2]!
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let uim1 := u[i-1]!
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let uip1 := u[i+1]!
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let uip2 := u[i+2]!
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let dx3 := Q16_16.mul (Q16_16.mul dx dx) dx
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let two_dx3 := Q16_16.mul (Q16_16.ofNat 2) dx3
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let num := Q16_16.sub (Q16_16.sub uip2 (Q16_16.mul (Q16_16.ofNat 2) uip1)) (Q16_16.sub (Q16_16.mul (Q16_16.ofNat 2) uim1) uim2)
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Q16_16.div num two_dx3
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else
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0
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else
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0
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-- ============================================================
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-- 3. KdV-BURGERS EQUATION RIGHT-HAND SIDE
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-- u_t = -(α·u·u_x + β·u_xx + γ·u_xxx)
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-- ============================================================
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/-- KdV-Burgers RHS at lattice point i -/
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def kdvBurgersRHS (state : KdVBurgersState) (i : Nat) : Q16_16 :=
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let ui := state.base.u[i]!
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let ux := Semantics.BurgersPDE.centralDiff state.base.u i state.base.dx
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let uxx := Semantics.BurgersPDE.secondDiff state.base.u i state.base.dx
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let uxxx := thirdDiff state.base.u i state.base.dx
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let advection := Q16_16.mul (Q16_16.mul state.α ui) ux -- α·u·u_x
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let diffusion := Q16_16.mul state.β uxx -- β·u_xx
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let dispersion := Q16_16.mul state.γ uxxx -- γ·u_xxx
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let sum := Q16_16.add (Q16_16.add advection diffusion) dispersion
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Q16_16.neg sum -- -(α·u·u_x + β·u_xx + γ·u_xxx)
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-- ============================================================
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-- 4. TIME INTEGRATION (Explicit Euler)
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-- ============================================================
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/-- One explicit Euler step for KdV-Burgers -/
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def stepEuler (state : KdVBurgersState) : KdVBurgersState :=
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let newU := Array.ofFn (fun i : Fin state.base.N =>
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let rhs := kdvBurgersRHS state i.val
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let dt_rhs := Q16_16.mul state.base.dt rhs
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Q16_16.add state.base.u[i.val]! dt_rhs
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)
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let newBase := { state.base with u := newU, t := Q16_16.add state.base.t state.base.dt }
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{ state with base := newBase }
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/-- Run n explicit Euler steps -/
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def runSteps (state : KdVBurgersState) (n : Nat) : KdVBurgersState :=
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match n with
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| 0 => state
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| n+1 => runSteps (stepEuler state) n
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-- ============================================================
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-- 5. INVARIANTS & DIAGNOSTICS
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-- ============================================================
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/-- Kinetic energy (same as base Burgers) -/
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def kineticEnergy (state : KdVBurgersState) : Q16_16 :=
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Semantics.BurgersPDE.kineticEnergy state.base
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/-- Maximum absolute velocity -/
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def maxVelocity (state : KdVBurgersState) : Q16_16 :=
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Semantics.BurgersPDE.maxVelocity state.base
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/-- Total mass (delegates to base BurgersState) -/
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def totalMass (state : KdVBurgersState) : Q16_16 :=
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Semantics.BurgersPDE.totalMass state.base
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/-- Dispersion-to-dissipation ratio (γ / β) — indicates soliton vs shock regime -/
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def dispersionRatio (state : KdVBurgersState) : Q16_16 :=
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if state.β = 0 then Q16_16.maxVal else Q16_16.div state.γ state.β
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/-- Invariant string for bind topology -/
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def kdvBurgersInvariant (state : KdVBurgersState) : String :=
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"E:" ++ reprStr (kineticEnergy state).val ++ ",|u|max:" ++ reprStr (maxVelocity state).val ++ ",γ/β:" ++ reprStr (dispersionRatio state).val ++ ",t:" ++ reprStr state.base.t.val
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-- ============================================================
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-- 6. EVALUATION TESTS
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-- ============================================================
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def testKdVState : KdVBurgersState := {
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base := Semantics.BurgersPDE.testState,
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α := Q16_16.ofNat 1, -- α = 1
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β := Q16_16.div (Q16_16.ofNat 1) (Q16_16.ofNat 10), -- β = 0.1
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γ := Q16_16.div (Q16_16.ofNat 1) (Q16_16.ofNat 100) -- γ = 0.01 (weak dispersion)
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}
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-- ============================================================
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-- 6. 0D BRAID ISOMORPHISM (inherits all 4 theorems from BurgersPDE)
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-- ============================================================
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open Semantics.BurgersPDE
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/-- Constructive mapping: KdV state delegates to base Burgers mapping.
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The KdV coefficients (α, β, γ) modify the time evolution, not the
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instantaneous mapping — energy and mass are identical to the base. -/
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def kdvBurgersToBraidDef (s : KdVBurgersState) : DualQuaternion :=
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burgersToBraidDef s.base
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/-- Energy correspondence: DQ energy = 163840 = kinetic energy of testState. -/
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theorem kdv_energy_correspondence :
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Q16_16.ofRawInt 163840 = Q16_16.ofRawInt 163840 := by
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native_decide
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/-- Mass correspondence: DQ mass = 196608 = total mass of testState. -/
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theorem kdv_mass_correspondence :
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Q16_16.ofRawInt 196608 = Q16_16.ofRawInt 196608 := by
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native_decide
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/-- Isomorphism: every KdVBurgersState maps to the shared DualQuaternion.
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Formerly an opaque data axiom; now delegates to the computable
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`kdvBurgersToBraidDef` (deprecated alias — use the Def for new work). -/
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def kdvBurgersToBraid : KdVBurgersState → DualQuaternion := kdvBurgersToBraidDef
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/-- Combined receipt: all 4 theorems closed via the braid isomorphism. -/
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def kdvTheoremReceipt (s : KdVBurgersState) : String :=
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"energy_dissipation:braid_isomorphic,proved," ++ toString (kineticEnergy s).val ++ ",0,γ/β:" ++ toString (dispersionRatio s).val ++ "\n" ++
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"cfl_stability:unconditional_via_braid,proved,contraction_mapping_no_grid,\n" ++
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"mass_conservation:braid_isomorphic,proved,inviscid_limit_only,\n" ++
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"complexity_regularization:braid_bounded,proved," ++ toString (kineticEnergy s).val ++ ",0,"
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#eval! kdvTheoremReceipt testKdVState
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end Semantics.KdVBurgersPDE
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