Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

129

while for the boundary m − 1 the flux is determined from the equation (4.36) F n + 1 m − 1 = F n + 1 m − 1 / 2 − A + m − 1 ( q n + 1 m − 1 / 2 − q n + 1 m − 3 / 2 ) .

(4.49)

Based on the equations (4.48) and (4.49) it follows F n + 1 m − F n + 1 m − 1 = F n m − F n m − 1 + A − m ∆ q n

m + 1 / 2 − A − m ∆ q n

m − 1 / 2 +

(4.50)

∆ q n

∆ q n

+ A +

A +

m − 1 / 2 −

m − 3 / 2 .

m − 1

m − 1

With the known flux vector F , Jacobi matrix A = ∂ F / ∂ q becomes

∂ F ∂ q =  

  , (4.51)

0

1

0

( γ − 3

2

− ( γ − 3 ) u 3 ( γ − 1 ) 2 t −

( γ − 1 )

2 ) u

A =

3 γ e

2

− γ ue t +( γ − 1 ) u

γ u

u

while Jacobi matrix ∂ H / ∂ q is defined by relation

dS dx  

0 0 0 1 2 u 2 − u 1 0 0 0   .

∂ H ∂ q = (

γ − 1 )

B =

(4.52)

Averaged variables 3 in the expressions (4.38) are introduced so that the equation F D − F L = A¯¯ ( q D , q L ) ( q D − q L ) is satisfied if the elements of the matrix A¯¯ are obtained by substitution of the independent flow variables in the corresponding elements of matrix A with such averaged quantities. Matrix A in the expression (4.51) can be diagonalized into form A = T Λ T − 1 , (4.54) where Λ is a diagonal matrix whose diagonal elements are eigenvalues of the matrix A . T is a matrix whose rows are right eigenvectors of the matrix A , ie. T =   1 α α u α ( u + c ) α ( u − c ) 1 2 u 2 α ( 1 2 u 2 + uc + c 2 γ − 1 ) α ( 1 2 u 2 − uc + c 2 γ − 1 )   , (4.55) while T − 1 is the matrix whose rows are left eigenvectors of the matrix A T − 1 =    1 − 1 2 u 2 γ − 1 c 2 ( γ − 1 ) u c 2 − ( γ − 1 ) c 2 β [( γ − 1 ) u 2 2 − uc ] β [ c − ( γ − 1 ) u ] β ( γ − 1 ) β [( γ − 1 ) u 2 2 + uc ] − β [ c +( γ − 1 ) u ] β ( γ − 1 )    . (4.56) 3 For a detailed determination of averaged variables procedure it is necessary to consult the literature [ hirsch ] (Hirsch, C. Numerical Computation of Internal and External Flows , Vol. 2, pp. 463–467). (4.53)