Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

136

equations (4.74) can be written in a more compact form by introducing the vector of nonconservative variables V = ρ , u , p T ∂ V ∂ t + e A ∂ V ∂ x = e Q . (4.76) Jacobi transformation matrix from conservative into nonconservative variables is defined by the derivative M = ∂ U ∂ V . (4.77) Relation between Jacobi matrix A of conservative variables and matrix e A , associated with nonconservative variables , can be established using the matrix M and its inverse matrix M − 1 . Introduction of Jacobi matrix M into the equation (4.76) leads to the relation

∂ V ∂ t

∂ V ∂ x

+( M e AM − 1 ) M

= M e Q .

M

(4.78)

Comparing the equation (4.78) with the corresponding equation ∂ U ∂ t + A ∂ U ∂ x = Q

(4.79)

for conservative variables leads to relationships

e A = M − 1 AM and A = M e AM − 1 ,

(4.80)

whereby a source member can be written

e Q = M − 1 Q .

(4.81)

In the case of one-dimensional flow transformation matrix M has the form M =   1 · · u ρ · u 2 / 2 ρ u 1 / ( γ − 1 )   ,

(4.82)

while Jacobi matrices A and e A can be expressed by relations 0 1

  

   and

0

u 2 2

− ( 3 − γ )

3 − γ ) u

( γ − 1 )

(

A =

u 2 2

( γ − 1 ) u 3 − γ e

t u γ e t − 3 ( γ − 1 )

γ u

(4.83)

e A =   u ·

  .

ρ u

·

1 / ρ

· ρ c 2

u