Navier Sokes Problems

NUMERICAL METHODS FOR DIFFERENTIAL EQNS

NAVIER - STOKES

^∂_t (u(y) v) + ∇ · (u(y) v ⊗ v) + ∇p - ∇ · ϵ(v) + F = 0

• go to WEAK form, performing integrals piecewise

∫∫ Ω ∇· (u(y) v) q = 0

∫∫ Ω ϵ(v) : ∇ q = ∫∫ ∂Ω φ q = 0; this is already used to ensure div(u) = 0

(u(y) v · ∇) v = nonlinear term remains as it was

• Non-linear term ^∫_Ω (u · ∇v) vT is called a trilinear form

1. Usually we add a condition to Q : Q = L²(α) s.t. ∫ Q = 0
2. Algebraic counterpart of 1st eqn no time derivative
3. Iterative schemes for c(vⁿ, vⁿ, vⁿ) = F(v)^{∀ v ∈ H}_p

• Idea is to use at differential level a scheme like: initial guess for velocity u⁰; having discarded time it's only an iteration for nonlinearity
• Convergence strongly depends on choice of initial guess

[ A + C(v)][U] = [F] [ B ][ P ]

• Convergence strongly depends on properties of the flow

Recall of Newton method

E(X) = 0 → f_i(x₁, ..., x_n) = 0

x⁽⁰⁾

to line to F = 0, in intersection with x-axis

f' (x⁽⁰⁾) (x - x⁽⁰⁾) + f (x⁽⁰⁾) = 0

f' (x⁽⁰⁾) (x⁽¹⁾ - x⁽⁰⁾) + f (x⁽⁰⁾) = 0

f' (x⁽⁰⁾) (x⁽¹⁾) + f (x⁽⁰⁾) = 0

to be generalized to our case

DF (x⁽ⁱ⁾)

(x⁽ⁱ⁺¹⁾ = x⁽ⁱ⁾ + δx)

DF (x⁽ⁱ⁾) = [∂f_i/∂x_j … ∂f_i/∂x_n]⁽ⁱ⁾

J (x⁽ⁱ⁾)

we'll have to invert Jacobian matrix

J (x⁽ⁱ⁾) δx = -F (x⁽ⁱ⁾)

δx is the solution of the linear system

Stopping Criteria

• Residue: |P (x^(M))| must be as close as possible to zero
• f (x^(M)) = x^(M), choose a-priori tolerance ε

If |P (x^(M))|₂ l₂ ε , stop

F (x^(M)) = x^(M)

Increments (difference between 2 successive iterates: small enough)

In most cases both criterias are checked, to impose stronger requirements

In practical cases for our problems, residue is the most commonly used

What is DF (x) = J (x) in our case?

Consider a generic non-linear operator N (u)

DN (u⁺)

active d.o.f., it could be (ℓ^(M+1-M))

h direction

DN (u⁺) h = lim_ε→0 N (u⁺ + εB) - N (u⁺) / ε

(definition of directional derivative)

N (u ) is LINEAR: DN (u⁺) h = lim_ε→0 N (u⁺) -εEN (B) - N (u⁺)/ ε = N (B)¹

N (u⁺) = 0

(DN (u⁺) h_b = N (B)_b = ∂h)

N (u⁺) is NONLINEAR ( (u : ∇u) u )

DN (u⁺) h =

lim_ε→0 ((u⁺ + εB) . ∇ (u⁺ + εB) . (u⁺ + εB) - (u⁺ . ∇u⁺ ) u⁺) / ε

lim_ε→0 (u+h) . ∇(u+ε) u_b b + ε (E_b )b (u) + (u+h) . ∇ (D) u⁺

DN (u⁺) h₂ = (∇ (u⁺) h_b) h . (∇ (D) u⁺) h U⁺

-∇ E (u⁽ⁿ⁺¹⁾)

( u⁽ⁿ⁾ . ∇) u⁽ⁿ⁾ + (u⁽ⁿ⁺¹⁾ . ∇) u⁽ⁿ⁺¹⁾) g U⁺ + ∇p ( u⁽ⁿ⁺¹⁾) = f ( u⁽ⁿ⁾ ), V(f)

∇ . U⁽ⁿ⁺¹⁾ = 0

Newton methods for steady Navier-Stokes equations