Numerical Factorization of Propagation Operator for Hyperbolic Equations and Application to One-way, True Amplitude One-way Equations and Bremmer Series

Domain decomposition method based on One-Way approaches for sound propagation in a partially lined duct

“…In practice, we are using some information on the exact operator spectrum such as the branch position of evanescent modes or the range of convective modes branch, which are deduced from a local modal analysis. The choice of placement of β ± coefficients and the influence of their number on the method convergence are discussed at greater length in [20].…”

“…In this section, we revisit in a more theoretically framework the construction method of the One-Way equations using a purely numerical factorization that we have proposed in the seminal paper [20]. In the first part, a characterization technique of the right and left invariant spaces associated to the right and left-going modes, respectively, is presented.…”

“…In this way, we also have access to accurate information about the refraction and transmission phenomena, which occur in function of the complexity of the mean flow. More precisely, we can construct One-Way equations (True amplitude version or Bremmer series) which have a wider domain of validity [20] than classical formulations such as those proposed by Towne & Colonius. Furthermore, the accuracy of this information will be used in the next section to construct precise transmission conditions in the presence of discontinuities in the domain.…”

“…Finally, the numerical integration of the equations (43), (44), ( 45) and ( 46) is realized from a Discontinuous Galerkin scheme adapted to the first-order hyperbolic equation [24]. This approach leads to an unconditionally stable explicit resolution [20].…”

“…However, this method is limited by a hypothesis of slow variation in the baseflow along the propagation direction, which includes also any variation in the transverse boundary conditions. A derivation of these One-Way equations has been presented in [20] and applied to the problem of a partially lined duct in [21]. This method is based on the full numerical construction of the projection operators that are needed to take into account the reflection and transmission of waves across any variation.…”