Halving the Error in Second Order Adams-Bashforth Methods via a Simple Time Filter

Abstract

This paper presents the implementation and rigorous analysis of a simple time filter applied to the second order Adams-Bashforth family of explicit numerical integration schemes. Although the implementation is remarkably straightforward—requiring the modular addition of just a single line of code—the resulting mathematical benefits are substantial, making it highly attractive for legacy scientific codebases. By theoretically modeling the coupled system as a unified linear multistep method, we are able to apply standard stability frameworks to the modified scheme. Specifically, we verify numerical stability using the Jury stability criterion, ensuring that the roots of the characteristic polynomial remain within the unit circle for the desired parameter range. Furthermore, we perform a detailed local truncation error analysis. Our results demonstrate that the filter acts to dampen the parasitic computational mode and effectively halves the leading error coefficient compared to the unfiltered method. This provides a robust enhancement to the original algorithm, yielding superior accuracy with negligible computational cost, as it avoids the expensive function evaluations associated with higher-order or implicit methods.

Keywords

• Stability analysis
• Error analysis
• Adam Bashforth
• Time filter

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