Understanding Backward Differentiation Formulas: Techniques for Success

Backward Differentiation Formulas (BDF) are widely used in numerical analysis, particularly for solving ordinary differential equations (ODEs). As the order of the BDF increases, it has been observed that while the method can achieve higher accuracy, it may also lead to instability, especially when applied to partial differential equations (PDEs). Notably, this instability typically arises when the parameter (k) exceeds 7, with (k=5) often considered optimal in diffusion simulations.

A key aspect of BDF is the parameter (k), which represents the number of time points utilized for the backward difference. This definition is essential in electrochemistry, where it differs from computer science nomenclature, which uses (k) to denote the number of intervals between time points. This distinction is crucial for practitioners applying BDF in their work, and it is the electrochemical convention that will be adhered to throughout this discussion.

One of the challenges faced when implementing BDF is how to initiate the process, especially with higher-order formulas such as the 5-point method. Given the requirement for multiple previous time points, it becomes impossible to apply the formula for initial time points before a certain threshold. Several strategies have emerged to address this issue, the simplest being the "simple start," which assumes initial values at times (t<0) equal to the value at (t=0). This method, while not particularly accurate on its own, can be significantly improved by applying a correction technique developed by Feldberg, which shifts the time values of iterations to enhance accuracy.

Another approach to starting the BDF process is known as the "rational start." This method begins with a 2-point BDF to generate the first output, subsequently employing higher-order BDFs, incrementally increasing from 3-point to 4-point, until the desired (k) is achieved. This technique tends to yield better accuracy than the simple start without correction, as it systematically builds upon previous calculations.

For those seeking even higher accuracy, the Kimble & White (KW) method offers a sophisticated starting point for BDF iterations. Originally designed for solving PDEs, this approach computes several unknown points as a system of equations simultaneously. This method not only initiates the BDF process effectively but also maintains a higher order of accuracy across the data set, making it a valuable technique for complex numerical solutions.

In summary, understanding the intricacies of BDF and its various starting techniques is vital for practitioners in fields that rely on numerical methods for solving ODEs and PDEs. By leveraging approaches like the simple start, rational start, and KW method, one can navigate the challenges of BDF effectively and enhance the reliability of their computational results.