Understanding Time Shifts in Numerical Solutions of Differential Equations

Numerical solutions of differential equations can be complex, particularly when dealing with time-dependent variables. A common technique involves utilizing a four-point approximation to the derivative at each unknown time level. This approach is an application of the Kimble & White (KW) method, which is particularly useful when approximating the behavior of systems over time.

An interesting aspect of numerical methods is how errors can be characterized. Traditionally, errors are reported as the difference between the numerical approximation and the exact solution at each time interval. However, an alternative approach involves calculating the time at which each computed value is exact, resulting in what is termed a "time shift." This method reflects the inherent nature of many simulations, where time is not explicitly included in the equations, leading to a permanent shift in the sequence of calculated values.

Different numerical methods exhibit unique time shift characteristics. For instance, the Euler and backward implicit methods demonstrate a linearly increasing time shift, while higher-order methods like the trapezium method maintain a time shift close to zero. Specifically, the BDF (backward differentiation formula) method with a simple start reveals a tendency towards a time shift of (-\frac{1}{2}\delta t), which can be correlated with the Feldberg time correction.

To analyze the efficiency of various starting protocols for BDF, tests were conducted using a standard ordinary differential equation. The performance of different starting methods was evaluated against the iteration number, revealing that while the simple start produced the largest error, the KW method yielded minimal errors, appearing almost indistinguishable from the zero error line. Notably, incorporating a time correction with the simple start significantly improved its performance.

These findings underline the importance of selecting an appropriate method for numerical simulation in fields such as electrochemistry. While the KW method can provide highly accurate results, its implementation is complex and computationally intensive. In contrast, the simple start with a time correction strikes an optimal balance between accuracy and efficiency, making it a preferable choice for many simulations.