488
R. Ahrens et al.
t 1 can be computed. Then, these moments can be used for computing the right-hand
side for the next time instance and so on.
Agglomeration and breakage processes can be also incorporated into the framework of QMOM, e.g., see [48]. An extension of the QMOM, which does not compute
the moments, but directly the weights and abscissas, is the Direct Quadrature Method
of Moments (DQMOM), as proposed in [47]. It is also possible to simulate multivariate populations with QMOM, e.g., see [18].
2.3.4 On Our Experience with Some of the Methods
As already mentioned above, it was noted in [36] that the use of a stabilized scheme
for convection-diffusion equations, which does not suppress spurious oscillations
sufficiently, often leads to a blow up of the simulations. Only cutting off such oscillations appropriately led to stable simulations. However, such cut-off techniques lead
inevitably to violations of conservation properties. Moreover, in the same paper, it
was concluded that the use of upwind techniques led to completely smeared and practically useless results. A clear improvement of the quality of the numerical solutions
was observed in [38] by using a linear FEM-FCT scheme for the convection-diffusion
and transport equations in the PBS. Based on this experience, we have employed the
linear FEM-FCT scheme for solving the energy equation (1.2) and concentration
equation (1.3) in PBSs. Different numerical methods for the 4d population balance
equation were studied in [13]. The problem of interest was a turbulent air-droplet
flow in a segment of a wind tunnel, where Ω × Ω int was a tensor product domain
in 4d. In this situation, FDM approaches can be applied easily. Two kinds of linear
FEM-FCT schemes and an FDM ENO scheme were compared. It turned out that the
FDM ENO scheme was by far the most efficient approach, such that it was recommended for population balance equations on tensor product domains. This scheme
was also applied successfully for the simulation of a bi-variate population balance
in [40]. In [3], a direct discretization using the FDM ENO approach for the population balance equation (1.4) and an operator-splitting scheme were compared for
an axisymmetric problem. While the operator-splitting scheme converged faster to
a steady-state, the evolution of the transition was predicted more accurately by the
direct discretization.
In summary, up to the publication of [3], we could, on the one hand, identify accurate and efficient approaches for simulating PBSs that are given on tensor product
domains. Here, efficiency refers only to the differential operators in the population
balance equation (1.4). Efficient methods for the integral operators are a different
topic, which will be discussed in Sect. 3.2. But on the other hand, it is very complicated to extend our favorite approach, the direct discretization, to problems defined in
more general domains, which occur usually in applications. In this respect, we could
make decisive progress in the preceding years by employing and further developing
a stochastic method, which will be discussed in detail in Sect. 2.4.
R. Ahrens et al.
t 1 can be computed. Then, these moments can be used for computing the right-hand
side for the next time instance and so on.
Agglomeration and breakage processes can be also incorporated into the framework of QMOM, e.g., see [48]. An extension of the QMOM, which does not compute
the moments, but directly the weights and abscissas, is the Direct Quadrature Method
of Moments (DQMOM), as proposed in [47]. It is also possible to simulate multivariate populations with QMOM, e.g., see [18].
2.3.4 On Our Experience with Some of the Methods
As already mentioned above, it was noted in [36] that the use of a stabilized scheme
for convection-diffusion equations, which does not suppress spurious oscillations
sufficiently, often leads to a blow up of the simulations. Only cutting off such oscillations appropriately led to stable simulations. However, such cut-off techniques lead
inevitably to violations of conservation properties. Moreover, in the same paper, it
was concluded that the use of upwind techniques led to completely smeared and practically useless results. A clear improvement of the quality of the numerical solutions
was observed in [38] by using a linear FEM-FCT scheme for the convection-diffusion
and transport equations in the PBS. Based on this experience, we have employed the
linear FEM-FCT scheme for solving the energy equation (1.2) and concentration
equation (1.3) in PBSs. Different numerical methods for the 4d population balance
equation were studied in [13]. The problem of interest was a turbulent air-droplet
flow in a segment of a wind tunnel, where Ω × Ω int was a tensor product domain
in 4d. In this situation, FDM approaches can be applied easily. Two kinds of linear
FEM-FCT schemes and an FDM ENO scheme were compared. It turned out that the
FDM ENO scheme was by far the most efficient approach, such that it was recommended for population balance equations on tensor product domains. This scheme
was also applied successfully for the simulation of a bi-variate population balance
in [40]. In [3], a direct discretization using the FDM ENO approach for the population balance equation (1.4) and an operator-splitting scheme were compared for
an axisymmetric problem. While the operator-splitting scheme converged faster to
a steady-state, the evolution of the transition was predicted more accurately by the
direct discretization.
In summary, up to the publication of [3], we could, on the one hand, identify accurate and efficient approaches for simulating PBSs that are given on tensor product
domains. Here, efficiency refers only to the differential operators in the population
balance equation (1.4). Efficient methods for the integral operators are a different
topic, which will be discussed in Sect. 3.2. But on the other hand, it is very complicated to extend our favorite approach, the direct discretization, to problems defined in
more general domains, which occur usually in applications. In this respect, we could
make decisive progress in the preceding years by employing and further developing
a stochastic method, which will be discussed in detail in Sect. 2.4.