Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

134

Quasi-linear form of the equation (4.63) can be written as follows:

∂ U ∂ t

∂ U ∂ x

∂ U ∂ y

∂ U ∂ z

+ A

+ B

+ C

= 0 ,

(4.66)

where A , B and C are three Jacobi flux matrices, determined by derivatives

∂ f ∂ U ,

∂ g ∂ U

∂ h ∂ U .

A =

B =

i C =

(4.67)

The Jacobi matrix A can be explicitly expressed in the form

       

    ,

0

1

0

0

0

γ − 1 2

q 2

− u 2 +

( 3 − γ ) u

− ( γ − 1 ) v − ( γ − 1 ) w

( γ − 1 )

A =

− uv − uw

v

u 0

0 u

0 0

(4.68)

w

γ − 1 2

( q 2 + 2 u 2 ) − ( γ − 1 ) uv − ( γ − 1 ) uw

2 ] γ e

− u [ γ e t − ( γ − 1 ) q

γ u

t −

while the matrix B is

    ,

0

0 v

1 u

0 0

0 0

− uv

γ − 1 2

q 2

− v 2 +

− ( γ − 1 ) u

( 3 − γ ) v

− ( γ − 1 ) w ( γ − 1 )

B =

(4.69)

− vw

0

w

v

0

γ − 1 2

( q 2 + 2 v 2 ) − ( γ − 1 ) vw

2 ] − ( γ − 1 ) uv γ e

− v [ γ e t − ( γ − 1 ) q

γ v

t −

where the matrix C is

   

    .

0

0

0 0

1 u v

0 0 0

− uw − vw

w

0

w

C =

(4.70)

γ − 1 2

q 2

− w 2 +

− ( γ − 1 ) u

− ( γ − 1 ) v

( 3 − γ ) w

( γ − 1 )

γ − 1 2

( q 2 + 2 w 2 )

2 ] − ( γ − 1 ) uw − ( γ − 1 ) vw γ e t −

− w [ γ e t − ( γ − 1 ) q

γ w

In the expressions (4.68)-(4.70) the variable q represents the intensity of the fluid velocity. By introducing the vector of basic variables V = ( ρ , u , v , w , p ) T a system of Euler equations receives the form

∂ V ∂ t

∂ V ∂ x

∂ V ∂ y

∂ V ∂ z

+ e A

+ e C

+ e B

= 0 ,

(4.71)

where e A , e B and e C are the corresponding Jacobi matrices. Matrices ( e A , e B , e C ) can be