THESE de DOCTORAT - cerfacs

3.3 Solving the system Ax = b 47
[
Ai,j + B i,j
] ˆpo = C j ⇒ A i,j ˆp o = C j (3.15)
in which B i,j = 0. Finally, there is another boundary condition that imposes a relation between
the acoustic velocity normal to the surface û · n and the acoustic pressure ˆp. Injecting the
definition of the acoustic impedance Ẑ = ˆp/iωû · n into Eq. (3.14) leads to:
Ẑ∇ ˆp ·⃗n − iω ˆp = 0 (3.16)
This type of boundary condition is known as a ‘Robin’ boundary condition. Note that B i,j ̸= 0
as soon as Ẑ ̸= 0 and Ẑ ̸= ∞ over some piece of the boundaries of the flow domain.
3.3 Solving the system Ax = b
From Eq. (3.9) it can be seen that the algebraic system to solve is of the form Ax = b. Several
approaches can be used to solve linear systems; they are divided mainly in two groups: noniterative
or ‘direct’ methods and iterative methods. Within ‘direct’ methods, two approaches are
most often used: the LU (or Gauss Elimination) and the QR decomposition. Both of them have
the same philosophy: “divide and conquer". In mathematical terms it means
A = LU or A = QR (3.17)
where A is a n × n matrix, L is a lower triangular n × n matrix, U is an upper triangular n × n
matrix, Q is an orthonormal 3 n × n matrix and R is an upper triangular n × n matrix. When
solving the linear system by the LU decomposition, three steps are followed
1. Ax = b → LUx = b
2. solve Lw = b for the unknown w
3. solve Ux = w for the unknown x.
If a QR decomposition is preferred, three steps are also considered
1. Ax = b → QRx = b
2. solve Qw = b for the unknown w
3 An orthornormal matrix is a matrix with n orthogonal columns in which the norm of each column is equal to
one.