Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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If λ ( j ) is the eigenvalue of the matrix e A , ie. root of the equation det | λ I − e A | = 0 , then the left eigenvectors ˜ ℓ

(4.84)

( j ) , defined as row vectors in the three-dimensional space of the

vector V , are the solution of the equation ˜ ℓ ( j ) e A (4.85) From the equation (4.85) can be concluded that there exists a matrix L − 1 that diagonalizes matrix e A . By defining the matrix L − 1 , whose rows are left eigenvectors ˜ ℓ ( j ) , ie. j th row of matrix L − 1 is the left eigenvector ˜ ℓ ( j ) , the equation (4.85), after grouping all its eigenvectors, can be written in the form L − 1 e A = Λ L − 1 , (4.86) where Λ is the diagonal matrix of all eigenvalues of matrix e A Λ =   λ 1 0 0 0 λ 2 0 0 0 λ 3   . (4.87) Based on the equations (4.86) and (4.87) the relation between the Jacobi matrix e A and diagonal matrix Λ can be determined e A = L Λ L − 1 and Λ = L − 1 e AL . (4.88) By introducing the matrix L − 1 and its inverse L it is possible to write the compatibil ity equation in a more compact form after multiplication of the equation (4.76) by the matrix L − 1 ( L − 1 ∂ t + L − 1 e A ∂ x ) V = L − 1 e Q . (4.89) Equation (4.89) introduces a new set of characteristic variables . These variables are defined as a column vector W by the relation that is valid for an arbitrary variations δ δ W = L − 1 δ V . (4.90) The roots of the equation (4.84) determine eigenvalues of the matrix e A λ 1 = u , λ 2 = u + c and λ 3 = u − c , (4.91) while the three eigenvectors of the matrix e A form diagonalizing matrix, which after normalization becomes L − 1 =     , ie. L =     . (4.92) = λ ( j ) ˜ ℓ ( j ) .

1 0 − 1 / c 2 0 1 1 / ρ c 0 1 − 1 / ρ c

1 ρ / 2 c − ρ / 2 c 0 1 / 2 1 / 2 0 ρ c / 2 − ρ c / 2