Understanding Oscillations in Crank-Nicolson Method Simulations

The Crank-Nicolson (CN) method is a popular numerical technique used for solving differential equations, particularly in the field of electrochemistry. However, it has a significant drawback: under certain initial conditions, particularly sharp changes in concentration, CN can generate persistent oscillations around zero. These oscillations can complicate simulations, particularly when high λ values are involved, leading many simulators to explore alternative methods.

The oscillatory behavior of the CN method becomes apparent in practical applications, such as the Cottrell system simulations. Graphical comparisons reveal that while the Laasonen method appears smoother, it can yield a higher relative error by the end of the simulation period. This makes it essential to understand the oscillation phenomenon in CN to make informed decisions when selecting numerical methods.

To mitigate the oscillation issue in CN simulations, one effective approach is to dampen these oscillations by adjusting the λ values. A λ value greater than 0.5 typically leads to the oscillatory response, but by reducing λ—usually by decreasing the time step (δT)—oscillations can be effectively controlled. Although this method can extend execution times, the good news is that once these oscillations are damped within the initial time interval, they are unlikely to recur.

One innovative strategy for achieving this damping involves subdividing the first time interval into smaller segments. This can be done using equal intervals or exponentially expanding intervals, known as the Pearson method. Numerical experiments suggest that a sub-λ value close to unity during these subdivisions is sufficient to minimize oscillations. This approach not only simplifies the computation but also maintains equal time intervals, which can be beneficial in various simulations.

The choice between using evenly spaced or exponentially expanding intervals often comes down to personal preference and the specific requirements of the simulation. By carefully selecting the method and adjusting the parameters, researchers can enhance the performance of the CN method while addressing its inherent oscillatory issues.