Pn1 pn At [bi2F12pnF12pniB12pn1 1252

Euler's method is called an explicit method because p^1 can be written explicitly in terms of pj. Each time we use the above technique to update pj, we introduce a small error due to the finite size of At and round-off error. In the absence of round-off error, we can achieve any desired accuracy by decreasing At. However, there are some problems with this approach. Usually, the biggest problem is the amount of computer time required when we choose a very small At. However, the round-off error incurred in each step does not decrease with At; rather it accumulates. It is possible that if At is too small, the total error is dominated by the round-off error. In that situation, the more steps we take, the larger the accumulated error. So a careful choice of time step is important.

Numerical stability is another issue with which we have to contend. That is, we do not want our numerical solutions to run off to when the real solution is bounded for all time. It is possible to show that Euler's method is stable only if

where max in the above equation means to use the value of n that produces the largest value of the quantity in the parentheses. Figure 12.9 illustrates this change in stability by using time steps slightly above and below Atc. To get an intuitive feel for this

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Figure 12.9 Numerical stability/instability. (A) When the time step is slightly below the critical step size, the numerical solution is stable. (B) When the time step is slightly above the critical step size, the numerical solution is unstable.

instability, let us consider one time step of the numerical scheme. At t = 0, we take pm = 1 and p0n = 0 for n = m. From (12.52) we have pm = [i - At(B„_1/2+Fm+1/2)j pm (12.54)

It is clear that if At > Atc, then pm will be negative. Since pm is a probability, negative values clearly do not make sense. The condition on At for stability is rather restrictive. Using the jump rates given in (12.84) and (12.85), it is possible to show that

This implies that in order to reduce the spatial step by a factor of 10 (which could be necessary to model accurately spatial fluctuations of the force, for example), the time step must be reduced by a factor of 100.

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