Euler Backward Iterative (EBI) and Modified Euler

Backward Iterative (MEBI) solvers

The EBI solver developed by Hertel et al. [89] uses

functional iteration to obtain a solution to the implicit Euler

backward approximation. To speed up convergence, groups

of species that are strongly coupled are isolated from the rest

of the species in the mechanism and analytical expressions

are derived to compute their concentrations. These results

are then used with the backward Euler approximation to

obtain estimates of the concentrations for the remainder of

the mechanism species. The concentrations of the group

species and the Euler backward species are updated using

the results from the previous iteration until convergence is

achieved. Hertel et al. [89] applied this technique to a

particular version of the CB4 mechanism and found it to be

more efficient than a version of the QSSA that employed

species lumping. They point out that the procedure applies

to the specific mechanism described, and may need to be

modified when implemented for other mechanisms.

In CMAQ, two variants of the EBI approach are included.

First, the same approach used by Hertel et al. has been

applied to the CMAQ CB4 mechanism. Analytical

expressions have been derived for four different groups of

strongly coupled species: 1) NO, NO2, O3, and O(3P); 2)

OH, HO2, HONO, and HNO4; and 3) NO3 and N2O5, and 4)

PAN and C2O3. The EBI iterative method is then used for

the remaining species. The second variant (Modified EBI or

MEBI) utilizes the method of Huang and Chang [90] to

replace the analytical expressions for groups 1 and 2 with

Newton-Raphson numerical solutions (analytical

expressions are retained for groups 3 and 4). This approach

is slower than the EBI approach because it requires matrix

inversions, but it is somewhat more amenable to

generalization. Even though the MEBI is slower than EBI,

it is still significantly faster than either SMVGEAR or

QSSA on scalar computers. To date, the MEBI approach has

been developed for the following CMAQ mechanisms –

CB4, SAPRC99, and the RADM2 variant utilizing the 4-

product isoprene chemistry. The development of EBI

solvers for other CMAQ mechanisms will be considered for

future versions of CMAQ.



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