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Unformatted text preview: 7.1. ISOTHERMAL, ISOCHORIC KINETICS 265 – easy to program, since Eq. (7.255) can be solved explicitly to predict the new value ρ n +1 in terms of the old values at step n . – need to have Δ t < τ fastest 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. (7.253) as follows: ρ n +1 − ρ n Δ t = f ( ρ n +1 ) , (7.256) so that ρ n +1 = ρ n + Δ t f ( ρ n +1 ) . (7.257) Implicit methods are summarized as – more difficult to program since a non-linear set of algebraic equations, Eq. (7.257), 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 < Δ t , – in general better performers than explicit methods.in general better performers than explicit methods....
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This note was uploaded on 11/26/2011 for the course EGN 3381 taught by Professor Park-sou during the Fall '11 term at FSU.
- Fall '11