AerE 546 Lecture 16 (Review finite-diff. and metrics)

AerE 546 ' Lecture 16


(Review finite-diff. and metrics)

Second derivatives: Evaluate ∂2ψ/∂2x as ∂xF with F = ∂xψ
Outside, then in, but iterated central would give 5 i-points.

Wrong:
δξ(F) =( Fi+1 - Fi-1 )/2 with F= δξψ → [(ψi+2 -ψi) - (ψi - ψi-2 )]/4.
Instead use:
( Fi+1/2 - Fi-1/2 )/2 → [(ψi+1 -ψi) - (ψi - ψi-1 )]/4.
Fi+1/2 = value on cell face Fi+1 is data at cell center.

Or recall formulating equations in conservation form: Flux in - Flux out + source = rate of
change inside c.v. Recall 1-D diffusion

∂t T = ∂x (κ∂xT) = -∂xF where F = -κ∂xT
n grid n

i-1/2 ← Δx → i+1/2' ' Flux in is in -n direction.

∂t ∫Tdx = - (F·n)i-1/2 - (F·n) i+1/2 = Fi-1/2 - Fi+1/2 is exact conservation.




This study source was downloaded by 100000899610689 from CourseHero.com on 09-06-2025 01:30:59 GMT -05:00
1
https://www.coursehero.com/file/33324864/Lecture16pdf/

, AerE 546 ' Lecture 16


Finite difference method for curvilinear grids

We have a method for chain rule derivatives:
Rtxy(*,1,1) = ∂ξ/∂x ; Rtxy(*,1,2) = ∂ξ/∂y
Rtxy(*,2,1) = ∂η/∂x ; Rtxy(*,2,2) = ∂η/∂y

e.g., grid, x(i,j), y(i,j) → Rtxy(*,1,1) = δy/δj / (δx/δi·δy/δj - δx/δj·δy/δi)

∂ψ/∂x = ∂ξ/∂x ∂ψ/∂ξ + ∂η/∂x ∂ψ/∂η = Rtxy(i,j,1,1) [ψ(i+1,j) - ψ(i-1,j)]/2
' ' ' ' ' + Rtxy(i,j,2,1) [ψ(i,j+1) - ψ(i,j-1)]/2

Use this to discretize Poisson equation by centered finite differences.
Second
AerE derivatives
546 make it `messy’; chain rule leads to cross derivatives ∂2ψ/∂ ξ∂η
Lecture 25
(explain how this -> 9 pointSOR
stencil)for ∇2 ψ = 0 on curvilinear grid

Gauss-Seidel in ∆-form

Loop over i, j
9 point stencil
! "
δ2 δ2 # $% &
R=ω− + ψ (1)
δ2x δ2y 7 8 9
! ! !
∆ψ = R(i, j)/A5 (2) 4 5 6
! ! ! ✻η ↔j
ψ n+1 (i, j) = ψ n (i, j) + λ∆ψ (3) 1 2 3
! ! ! ✲
endloop ξ↔i
R is evaluated with the current values of ψ — a combination of n and n + 1.

General geometry: From
∂F ∂ξ ∂F ∂η ∂F ∂G ∂ξ ∂G ∂η ∂G
= + and = +
∂x ∂x ∂ξ ∂x ∂η ∂y ∂y ∂ξ ∂y ∂η
with F = ∂x ψ and G = ∂y ψ, the
O-cutLaplacian becomes

∂x ψ
# $% &
! ! " "
∂ξ ∂ ∂ξ ∂ψ ∂η ∂ψ ∂η ∂ ∂ξ ∂ψ ∂η ∂ψ
R = + + +
∂x ∂ξ ∂x ∂ξ ∂x ∂η ∂x ∂η ∂x ∂ξ ∂x ∂η (4)
! " ! "
∂ξ ∂ ∂ξ ∂ψ ∂η ∂ψ ∂η ∂ ∂ξ ∂ψ ∂η ∂ψ
So R can be+evaluated in aggregate.
+ A5 is
+needed too. +
∂y ∂ξ ∂y ∂ξ ∂y ∂η ∂y ∂η ∂y ∂ξ ∂y ∂η


This study source was downloaded by 100000899610689 from CourseHero.com on 09-06-2025 01:30:59 GMT -05:00
2
https://www.coursehero.com/file/33324864/Lecture16pdf/