Making Fluids Incompressible - Fluid Simulation for Computer Graphics - page 66

Let's take the derivative of the change in energy with respect to pressure
p i,j,k and set it to zero:

∂KE n +1
∂p i,j,k −

∂KE
∂p i,j,k

∂W
∂p i,j,k

+ =0 .

Pressure doesn't appear in the middle term, the intermediate kinetic en-
ergy, so that term drops out. Referring back to the discrete sums above,
we get the following gigantic equation:

u i +1 / 2 ,j,k −

Δ t
ρ i +1 / 2 ,j,k Δ x

Δ t
ρ i +1 / 2 ,j,k

p i +1 ,j,k − p i,j,k
Δ x

ρ i +1 / 2 ,j,k V i +1 / 2 ,j,k

u i− 1 / 2 ,j,k −

Δ t
ρ i− 1 / 2 ,j,k Δ x

Δ t
ρ i− 1 / 2 ,j,k

p i,j,k − p i− 1 ,j,k
Δ x

− ρ i− 1 / 2 ,j,k V i− 1 / 2 ,j,k

v i,j +1 / 2 ,k −

Δ t
ρ i,j +1 / 2 ,k Δ x

Δ t
ρ i,j +1 / 2 ,k

p i,j +1 ,k − p i,j,k
Δ x

+ ρ i,j +1 / 2 ,k V i,j +1 / 2 ,k

v i,j− 1 / 2 ,k −

Δ t
ρ i,j− 1 / 2 ,k Δ x

Δ t
ρ i,j− 1 / 2 ,k

p i,j,k − p i,j− 1 ,k
Δ x

− ρ i,j− 1 / 2 ,k V i,j− 1 / 2 ,k

w i,j,k +1 / 2 −

Δ t
ρ i,j,k +1 / 2 Δ x

Δ t
ρ i,j,k +1 / 2

p i,j,k +1 − p i,j,k
Δ x

+ ρ i,j,k +1 / 2 V i,j,k +1 / 2

w i,j,k− 1 / 2 −

Δ t
ρ i,j,k− 1 / 2 Δ x

Δ t
ρ i,j,k− 1 / 2

p i,j,k − p i,j,k− 1
Δ x

− ρ i,j,k− 1 / 2 V i,j,k− 1 / 2

Δ t
Δ x u solid

Δ t
Δ x u solid

− V i +1 / 2 ,j,k + V i− 1 / 2 ,j,k

i +1 / 2 ,j,k i− 1 / 2 ,j,k

Δ t
Δ x v solid

Δ t
Δ x v solid

− V i,j +1 / 2 ,k + V i,j− 1 / 2 ,k

i,j +1 / 2 ,k i,j− 1 / 2 ,k

− V i,j,k +1 / 2 Δ t

i,j,k +1 / 2 + V i,j,k− 1 / 2 Δ t

Δ x w solid Δ x w solid

i,j,k− 1 / 2

u solid

Δ t
Δ x

i +1 / 2 ,j,k − u solid

+ V i,j,k

i +1 / 2 ,j,k

i,j,k, +1 / 2 =0 .

+ v solid

i,j +1 / 2 ,k − v solid i,j +1 / 2 ,k w solid

− w solid

i,j,k +1 / 2