Частично линеаризованный неявный метод (метод Шимазаки)

Вычислительную часть неявного метода можно существенно упростить, если линеаризовать только разрушение относительно вычисляемого газа, предварительно разбив скорость фотохимического изменения на продукцию и разрушение.

Тогда система уравнений приобретает диагональный вид, а решение для каждого газа – простой вид

Однако опять при однократном применении этого метода может нарушиться баланс массы (консервативность), поэтому необходимо применять итерации

Полуаналитический метод решения уравнений химической кинетики

Рассмотрим линеаризованную обратную формулу Эйлера:

,

в которой компоненты уравнения химической кинетики, относящиеся ко всем другим газам берутся с предыдущего временного шага, а относящиеся к вычисляемому газу с текущего временного шага.

Если рассмотреть дифференциал от правой части, то он равен с учетом постоянства компонентов и

Тогда дифференциал выражается как

Если подставить это выражение в линеаризованную обратную формулу Эйлера, то получим

Разделим обе части на и получим пригодную для интегрирования формулу

Используя правила интегрирования, получаем

Откуда

Выражая отсюда текущее значение , получим

Или

Или

Когда скорость разрушения обращается в нуль, в знаменателе второго слагаемого правой части образуется неопределенность. Для этого случая необходимо определять условно другое решение

Если наоборот продукция обращается в нуль, то остается только первое слагаемое – аналог точного решения.

На практике условие разделения использования различных методов заключается в сравнении произведения с некоей пороговой величиной. Если значение этого произведения будет меньше 0.01, то газы могут рассматриваться как долгоживущие и их концентрации могут рассчитываться с использованием направленного вперед явного метода Эйлера

Если значение этого произведения больше 10, то газы являются короткоживущими и их концентрации могут рассчитываться из условия фотохимического равновесия

В промежутке между 0.01 и 10 можно использовать базовую формулу полуаналитического метода.

Multistep implicit–explicit (mie) solution to odes

A positive-definite, mass-conserving, unconditionally stable iterative technique that takes advantage of the forward and backward Euler methods is the multistep implicit–explicit (MIE) method (Jacobson 1994; Jacobson and Turco 1994). With this method, concentrations are estimated with an iterated backward Euler calculation, and the estimates are applied to reaction rates used in a forward Euler calculation of final concentrations. Upon iteration the forward Euler converges to the backward Euler. Since backward Euler solutions are always positive, forward Euler solutions must converge to positive values as well. A technique related to the MIE method is merely iterating the backward Euler equation until convergence occurs for all species. Such a method requires many more iterations than does the MIE method.

The steps for determining final concentrations with the MIE method are described below. To illustrate, four species, NO, NO2, O3, and O, and two reactions,

·NO + O3 → ·NO2 + O2 (k1)

O3 + hν → O2 + ·O· (J )

are considered. The change in O2 concentration by these reactions is ignored for the illustration.

The first step in the MIE solution is to initialize backward Euler concentrations (molec. cm−3) and maximum backward Euler concentrations for each active species i = 1, . . . , K with concentrations from the beginning of the simulation or, in the case of a new time step, with concentrations from the end of the last time step.

Thus,

Ni,B,1 = Ni,t−h

Ni,MAX,1 = Ni,t−h

where the subscript B indicates a backward Euler concentration, and the subscript 1 indicates the first iteration of a new time step. In subsequent equations, the iteration number is denoted by m. Here, m = 1. A maximum backward Euler concentration is required to prevent backward Euler concentrations from blowing up to large values upon iteration and is updated each iteration, as discussed shortly.

The second step is to estimate reaction rates by multiplying rate coefficients by backward Euler concentrations. Examples of two-body and photolysis reaction rates are

respectively, where Rc is the rate of reaction (molec. cm−3 s−1), n is the reactionrate number, m is the iteration number (= 1 for the first iteration), kn is the kinetic rate coefficient of the nth reaction, and Jn is the photolysis rate coefficient of the nth reaction. In the two-reaction example, the backward Euler rates are

The third step is to estimate production rates, loss rates, and implicit loss coefficients for each species from the reaction rates just calculated. The backward Euler production rate (molec. cm−3 s−1) of species i is

where Nprod,i is the number of reactions in which species i is produced, and Rc,nP (l,i ),B,m is the lth backward Euler production rate of species i. The array nP(l, i ) gives the reaction number corresponding to the lth production term of species i. In the two-reaction example, the summed production rates of O and NO2 are

respectively, where the two active species produced have only one production term each.

The backward Euler loss rate (molec. cm−3 s−1) of a species is

where Nloss,i is the number of reactions in which species i is lost, and Rc,nL(l,i ),B,m is the lth backward Euler loss rate of species i. The array nL(l, i ) gives the reaction number corresponding to the lth loss term of species i. In the two-reaction example, the summed loss rates of NO and O3 are

The backward Euler implicit loss coefficient (s−1) of a species is

In the two-reaction example, the implicit loss coefficients of NO and O3 are

Since a reactant concentration can equal zero, computing implicit loss coefficients with (12.57) can result in a division by zero. If the coefficients are, instead, computed directly from (12.58), division by zero is avoided.

The fourth step is to calculate backward Euler concentrations for all species at iteration m + 1 with

)

Such estimates are used to calculate production and loss terms during the next iteration.

The fifth step is to calculate forward Euler concentrations for all species at iteration m + 1 with

The sixth step is to check convergence. Convergence is determined by first checking, at the end of each iteration during a time step, whether all forward Euler concentrations from (12.60) exceed or equal zero. If they do, a counter, nP, initialized to zero before the first iteration of the time step, is incremented by one and another iteration is solved. If a single forward Euler concentration falls below zero during a given iteration, the counter nP is reset to zero, and a new iteration is solved to try to update nP again. Thus, whether the counter is updated or reset to zero is

determined by the following criteria:

ˆN

where ˆN F,m+1 is the entire set of forward Euler concentrations and Ni,F,m+1 is an individual concentration. If all forward Euler concentrations ≥ 0 for NP iterations in a row (e.g., if nP = NP), convergence is said to have occurred. NP is a constant that depends on the number of reactions and their stiffness. Typical values are 5 for large sets of equations and 30–50 for small sets of equations.

For faster solutions, the criteria can be modified so that, when h_c,i,B,m ≥ LT for a species during an iteration, the forward Euler concentration of the species does not need to exceed zero for nP to avoid being reset to zero. In such cases, the species is short-lived, and the backward Euler solution is more accurate than the forward Euler solution. LT is a constant between 102 and 106. Values of 102 speed solutions but increase errors. Values of 106 slow solutions but decrease errors. In sum, the following criteria identify conditions under which the counter is updated

or reset to zero under the modified convergence method:

With the modified method, convergence is again obtained when nP = NP. When the modified criterion is met, the final concentrations are

If the modified convergence criterion is not met, iterations continue. Before iterations continue, a seventh step, to update maximum backward Euler concentrations and to limit current backward Euler concentrations, is required.

Maximum backward Euler concentrations are limited to the larger of the current backward Euler concentration and the initial concentration from the time step. Current backward Euler concentrations are then limited to the smaller of their current value and the maximum concentration from the last iteration. In other words,

for each species. Note that the value of Ni,MAX used in (12.65) is from the previous iteration, while that calculated in (12.64) is from the current iteration. After these updates, the iteration returns to (12.51) until convergence is obtained. Here, it is shown that iterated backward Euler solutions converge to iterated forward Euler solutions and positive numbers. At the end of any iteration, the backward Euler concentration is

which is simply (12.59) rearranged. Equation (12.66) must be positive, since (12.59) cannot be negative. At the end of the same iteration, the forward Euler Yields

where _c,i,B,mNi,B,m = Lc,i,B,m. For the forward and backward Euler solutions to converge to each other,

must be satisfied. This always occurs upon iteration of the backward Euler solution. Thus, when the backward Euler solution converges, the forward Euler solution must converge to the same value, and both must converge to positive numbers. Figure 12.1 shows convergence of forward Euler to backward Euler concentrations versus iteration number for four species. Figure 12.2 compares MIE simulation results with an exact solution for two species when a set of 92 ODEs is solved simultaneously.