Exploring the Stability of High-Order Methods in Ordinary Differential Equations

In the realm of numerical analysis, the stability of algorithms is paramount, especially when solving ordinary differential equations (ODEs). One intriguing approach to achieving stability is the use of the Kimble & White (KW) method, which modifies traditional finite difference schemes to enhance performance. The KW method generates a system of equations that, while initially unstable when used in time-marching, can be transformed into a stable system with the addition of a specific boundary term.

At its core, the KW method begins with a central difference formula, yielding a local error of (O(\delta t^3) and a global error of (O(\delta t^2). However, the challenge arises when the algorithm is applied sequentially over time, resulting in divergence of the solution. To address this, the KW scheme incorporates values at multiple points in time, allowing for the formulation of an equation system that captures the dynamics of the ODE across these points.

To fill the gap in the central difference approximation, particularly for the last time point, Nguyen & White recommend utilizing a backward differentiation formula (BDF). This BDF approach not only stabilizes the overall system but also ensures that the stability is retained throughout the computation, yielding solutions with a reliable accuracy of (O(\delta t^2).

The extension of the KW method by Kimble & White introduces higher-order accuracy through the use of a four-point central difference scheme. This development necessitates the application of asymmetric forms for the first and last points, while employing a central form for the intermediate points. Although employing a higher-order method can enhance accuracy, it also presents stability challenges when run as a time-marching algorithm. Just as before, the inclusion of BDF forms stabilizes the system, turning potential instability into a robust order of convergence.

When applied to systems of partial differential equations (PDEs), the KW method further demonstrates its versatility. By reducing the evaluation points in both time and space, the method manages to cope with the potentially massive matrices that arise in these scenarios. This approach leads to a practical application of the KW method, which serves effectively as a foundation for developing high-order starting values in BDF methods, thereby establishing a new paradigm in numerical solutions for both ODEs and PDEs.

The ongoing development and refinement of the KW method illustrate the dynamic nature of numerical methods in mathematical modeling. As researchers continue to explore innovative techniques for enhancing stability and accuracy, the KW method stands as a compelling example of how traditional approaches can be adapted to meet the demands of complex differential equations.