THESE de DOCTORAT - cerfacs

3
Development of a numerical tool for
combustion noise analysis, AVSP-f
Contents
3.1 Discretizing the Phillips’ equation . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Boundary Conditions in AVSP-f . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Solving the system Ax = b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1 On the Arnoldi algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 The least square problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.2 Dynamic preconditioning: the embedded GMRES . . . . . . . . . . . . . 56
It has been seen in section 2.3.4 that when considering the D’Alembert operator in the wave
equation, it is possible to obtain an explicit solution by means of Green’s functions. Notably,
this is the case for Lighthill’s equation in both time and frequency domain (Eq. 2.43 and 2.46).
On the contrary, Phillips analogy (Eq. 2.59) has no known analytical solutions since its wave
operator is much more complex. Even if this equation is simplified to Eq. (2.62) or Eq. (2.63), the
fact of having a wave operator that accounts for changes in the mean speed of sound prevents
deriving an explicit solution of this wave equation.
41