Mathematical Physics - Volume II - Numerical Methods

Chapter 5. Review of Development of the Smooth Particle Hydrodynamics (SPH) Method

168

As W is an even function, the terms containing odd powers of x ′ − x vanish. Neglecting second and higher order terms, which is consistent with the overall order of the method, gives: Z Ω W ( | x − x ′ | , h ) f ( x ′ ) ∂ g ( x ′ ) ∂ x ′ d x ′ = = f ( x ) ∂ g ( x ) ∂ x Z Ω W ( | x − x ′ | , h ) d x ′ = f ( x ) ∂ g ( x ) ∂ x . (5.16) Substituting ∂ g ( x ) ∂ x for ∂ g ( x ) ∂ x gives: f ( x ) ∂ g ( x ) ∂ x = f ( x ) Z Ω W ( | x − x ′ | , h ) ∂ g ( x ′ ) ∂ x ′ d x ′ . (5.17) Using the last relation in equations (5.11), (5.12) and (5.13) yields D ρ ( x ) D t = − ρ ( x ) Z Ω W ( | x − x ′ | , h ) ∇ · v ( x ′ ) d x ′ , (5.18) D v D t = Z Ω W ( | x − x ′ | , h ) ∇ · σ ( x ′ ) ρ ( x ′ ) ! d x ′ +

σ ( x ) [ ρ ( x )] 2 Z Ω σ ( x ) [ ρ ( x )] 2 Z Ω σ ( x ) · v ( x )

W ( | x − x ′ | , h ) ∇ ρ ( x ′ ) d x ′ ,

(5.19)

+

D E ( x )

D t

W ( | x − x ′ | , h ) ∇ ( ρ ( x ′ ) v ( x ′ )) d x ′ −

=

[ ρ ( x )] 2 Z Ω

W ( | x − x ′ | , h ) ∇ ( ρ ( x ′ )) d x ′ .

(5.20)

−

Note that all equations include kernel approximations of spatial derivatives: ⟨ ∇ f ( x ) ⟩ = Z Ω ∇ f ( x ′ ) W ( | x − x ′ | , h ) d x ′ .

(5.21)

Integrating by parts gives:

⟨ ∇ f ( x ) ⟩ = Z Ω

∇ ( W ( | x − x ′ | , h ) f ( x ′ )) d x ′ −

(5.22)

− Z Ω

f ( x ′ ) ∇ W ( | x − x ′ | , h ) d x ′ .

Using Green’s theorem, the first term of the right hand side can be rewritten as: Z Ω ∇ · f ( x ′ ) W ( | x − x ′ | , h ) d x ′ = Z Ω f ( x ′ ) W ( | x − x ′ | , h ) n d S = 0 .

(5.23)