3 SIM
1.1. ISOTHERMAL, ISOCHORIC KINETICS 51
@NOD
@ND
10-8 10-6 10-4 0.01 1 100
tHsL 10-16
10-14 10-12 10-10 10-8
@ND,@NOD HmoleccL
Figure 1.11: ρN O and ρN versus time for Zel’dovich mechanism at T = 1500 K, P = 6.2355×103 dyne/cm2.
the high pressureP = 6.2355×105 dyne/cm2, T = 1500K case, we notice some similarities and dramatic differences. The overall shape of the time-profiles of concentration variation is similar. But, we see the reaction commences at a much later time, t ∼ 10−6 s. For t ∼10−4 s, there is a temporary cessation of significant reaction. We notice a long plateau in which species concentrations do not change over several decades of time. This is again actually a pseudo-equilibrium. Significant reaction recommences for t∼ 10s. Only around t∼100sdoes the system approach final equilibrium. We can perform an eigenvalue analysis both at the initial state and at the equilibrium state to estimate the time scales of reaction.
At the initial state, we find
τ1 = 2.37×10−6 s, (1.245)
τ2 = 4.25×10−5 s. (1.246)
The onset of significant reaction is consistent with the prediction given by τ1 at the initial state. Moreover, initially, the reaction is not very stiff; the stiffness ratio is κ = 17.9.
Interestingly, by decreasing the initial pressure by a factor of 102, we increased the initial time scales by a complementary factor of 102; moreover, we did not alter the stiffness.
At equilibrium, we find
t→∞lim ρN O = 4.6×10−11 mole
cm3 , (1.247)
t→∞lim ρN = 4.2×10−16 mole
cm3 , (1.248)
(1.249)
and
τ1 = 7.86×10−5 s, (1.250)
τ2 = 3.02×101 s. (1.251)
By decreasing the initial pressure by a factor of 102, we decreased the equilibrium concentra- tions by a factor of 102 and increased the time scales by a factor of 102, leaving the stiffness ratio unchanged.
In summary, we find the effect of lowering the initial concentrations significantly while leaving temperature constant
• lowers the pressure significantly, proportionally slowing down the collision time, as well as the fastest and slowest time scales,
• does not affect the stiffness of the system.
1.1.2.2.3 Stiffness and numerics The issue of how to simulate stiff systems of ordinary differential equations, such as presented by our Zel’dovich mechanism, is challenging. Here a brief summary of some of the issues will be presented. The interested reader should consult the numerical literature for a full discussion. See for example the excellent text of Iserles. 9 We have seen throughout this section that there are two time scales at work, and they are often disparate. The species evolution is generally characterized by an initial fast transient, followed by a long plateau, then a final relaxation to equilibrium. We noted from the phase plane of Fig. 1.9 that the final relaxation to equilibrium (shown along the green line labeled
“SIM”) is an attracting manifold for a wide variety of initial conditions. The relaxation onto the SIM is fast, and the motion on the SIM to equilibrium is relatively slow.
Use of common numerical techniques can often mask or obscure the actual dynamics.
Numerical methods to solve systems of ordinary differential equations can be broadly cat- egorized as explicit or implicit. We give a brief synopsis of each class of method. We cast each as a method to solve a system of the form
dρ
dt =f(ρ). (1.252)
• Explicit: The simplest of these methods, the forward Euler method, discretizes Eq. (1.252) as follows:
ρn+1−ρn
∆t =f(ρn), (1.253)
so that
ρn+1 =ρn+ ∆t f(ρn). (1.254)
Explicit methods are summarized as
9A. Iserles, 2008, A First Course in the Numerical Analysis of Differential Equations, Cambridge Uni- versity Press, Cambridge, UK.
1.1. ISOTHERMAL, ISOCHORIC KINETICS 53 – easy to program, since Eq. (1.254) can be solved explicitly to predict the new
value ρn+1 in terms of the old values at step n.
– need to have ∆t < τf astest in order to remain numerically stable,
– able to capture all physics and all time scales at great computational expense for stiff problems,
– requiring much computational effort for little payoff in the SIM region of the phase plane, and thus
– inefficient for some portions of stiff calculations.
• Implicit: The simplest of these methods, the backward Euler method, discretizes Eq. (1.252) as follows:
ρn+1−ρn
∆t =f(ρn+1), (1.255)
so that
ρn+1 =ρn+ ∆t f(ρn+1). (1.256)
Implicit methods are summarized as
– more difficult to program since a non-linear set of algebraic equations, Eq. (1.256), must be solved at every time step with no guarantee of solution,
– requiring potentially significant computational time to advance each time step, – capable of using very large time steps and remaining numerically stable,
– suspect to missing physics that occur on small time scales τ <∆t, – in general better performers than explicit methods.
A wide variety of software tools exist to solve systems of ordinary differential equations.
Most of them use more sophisticated techniques than simple forward and backward Euler methods. One of the most powerful techniques is the use of error control. Here the user specifies how far in time to advance and the error that is able to be tolerated. The algorithm, which is complicated, selects then internal time steps, for either explicit or implicit methods, to achieve a solution within the error tolerance at the specified output time. A well known public domain algorithm with error control is provided by lsode.f, which can be found in the netlib repository. 10
Let us exercise the Zel’dovich mechanism under the conditions simulated in Fig. 1.11, T = 1500 K, P = 6.2355×103 dyne/cm2. Recall in this case the fastest time scale near equilibrium is τ1 = 7.86×10−5 s ∼10−4 s at the initial state, and the slowest time scale is
10 Hindmarsh, A. C., 1983,“ODEPACK, a Systematized Collection of ODE Solvers,” Scien- tific Computing, edited by R. S. Stepleman, et al., North-Holland, Amsterdam, pp. 55-64.
http://www.netlib.org/alliant/ode/prog/lsode.f
Explicit Explicit Implicit Implicit
∆t (s) Ninternal ∆tef f (s) Ninternal ∆tef f (s)
102 106 10−4 100 102
101 105 10−4 100 101
100 104 10−4 100 100
10−1 103 10−4 100 10−1
10−2 102 10−4 100 10−2
10−3 101 10−4 100 10−3
10−4 100 10−4 100 10−4
10−5 100 10−5 100 10−5
10−6 100 10−6 100 10−6
Table 1.3: Results from computing Zel’dovich NO production using implicit and explicit methods with error control in dlsode.f.
τ = 3.02×101 s at the final state. Let us solve for these conditions using dlsode.f, which uses internal time stepping for error control, in both an explicit and implicit mode. We specify a variety of values of ∆t and report typical values of number of internal time steps selected by dlsode.f, and the corresponding effective time step ∆tef f used for the problem, for both explicit and implicit methods, as reported in Table 1.3.
Obviously if output is requested using ∆t > 10−4 s, the early time dynamics near t ∼ 10−4swill be missed. For physically stable systems, codes such asdlsode.fwill still provide a correct solution at the later times. For physically unstable systems, such as might occur in turbulent flames, it is not clear that one can use large time steps and expect to have fidelity to the underlying equations. The reason is the physical instabilities may evolve on the same time scale as the fine scales which are overlooked by large ∆t.