University:
Massachusetts Institute of Technology
Course:
18.336 | Numerical Methods for Partial Differential Equations
Academic year:

2023
Views:

476

Pages:

3
Author:

Kane Terrell

18.336 spring 2009 lecture 24 Pseudospectral Methods Vorticity: w � = � × �u 1 2 � �u Re 1 ⇒ � × �ut + � × ((�u · �)�u) = � × (−�p) + � × ( �2�u) � �� Re �� =0 �ut + (�u · �)�u = −�p + �× and �2 commute ⇒ w � t +�u ·�w �= 1 2 �w � Re Navier-Stokes equations in vorticity formulation How to ﬁnd �u from w �? 2D w = vx − uy scalar Stream Function: ⊥ � ψ(�x) s.t. �u = � ψ, i.e. u = ψy v = −ψx � ⇒ �2 ψ = ψxx + ψyy = −vx + uy = −w Thus: w �2 ψ=−w � u=�⊥ ψ −→ ψ −→ �u Remark: Incompressibility guaranteed � · �u = ux + vy = ψyx − ψxy = 0 � Time Discretization Semi-implicit: • �u · �w explicit • �2 w Crank-Nicolson � wn+1 − wn 1 1� 2 n · � w + �2 wn+1 + (�un · �wn ) = � �� Re 2 Δt =N n � � � � 1 1 1 1 2 2 n+1 n ⇒ I− � w = −N + I+ � wn Δt 2Re Δt 2Re ↑ If discretized in space → linear system Here: want to use spectral methods 1 05/07/09 Fourier Transform Properties: � +∞ ˆ F (f ) = f (k) = f (x)e−2πikx dx −∞ � +∞ −1 ˆ F (f ) = f (x) = fˆ(k)e2πikx dk F (f � ) = 2πikF (f ) −∞ F (�2 f ) = −4π 2 |�k|2 F (f ) ← 2D FT (|�k|2 = kx 2 + ky 2 ) F (f ∗ g) = F (f ) · F (g) F (f · g) = F (f ) ∗ F (g) Use Here: � � � � 1 4π 2 � 2 1 4π 2 � 2 n+1 n � − + |k | ŵ = −N + |k | ŵn Δt 2Re Δt 2Re � � 2 1 2π 2 n � | ŵn ˆ + − −N |k Δt Re ⇒ ŵn+1 = 1 2π 2 � 2 |k| + Δt Re ˆ: Computation of N � N = �u · �w ⇒ N̂ = �uˆ ∗ �w Ineﬃcient in Fourier space. So perform multiplication in physical space. Numerical Method: Time step ŵn → ŵn+1 : 1 1. ψˆn = − wˆ n |�k| ⎧ n û = −2πiky ψ̂ n ⎪ ⎪ ⎨ n v̂ = 2πikx ψ̂ n 2. n ⎪ wˆ = −2πikx ŵn ⎪ ⎩ xn ŵy = −2πiky ŵn ⎧ n u ⎪ ⎪ ⎨ n v 3. iFFT → wx n ⎪ ⎪ ⎩ wy n 5. FFT → (N̂ n )∗ 6. Truncate kx , ky > 23 kmax to prevent aliasing � (N̂ )∗ kx , ky ≤ 23 kmax n N̂ = if 0 else 7. ŵn+1 = 4. N n = un · wx n + v n · wy n 2 � 1 2π 2 � 2 − |k | ŵn Δt Re 1 2π 2 � 2 |k| + Δt Re ˆ n+1 + −N � Particle Methods Linear � advection ut + cux = 0 u(x, 0) = u0 (x) Solution: u(x, t) = u0 (x − ct) Need to work hard to get good schemes for advection, on a ﬁxed grid. Reason: Represent sideways motion by up and down motion. Alternative: Move computed nodes, rather than having a ﬁxed grid. Image by MIT OpenCourseWare. Ex.: 1D Advection � ut + cux = 0 u(x, 0) = u0 (x) Particle method: 1. Sample initial conditions xj = jΔx, uj = u0 (xj ) 2. Move particles along characteristics: � � � ẋj = c xj (t) = xj (0) + ct ⇒ → Exact solution on particles. u̇j = 0 uj (t) = u0 (xj (0)) Image by MIT OpenCourseWare. Ex.: 2D Advection � φt + �v (�x) · �φ = 0 φ(�x, 0) = φ0 (�x) Particle method: 1. �xj , φj = φ0 (�xj ) � � �ẋj = �v (�xj ) 2. φ˙j = 0 Solve characteristic ODE, e.g. by RK4; very accurate on particles. 3

Pseudospectral Methods

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