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.
Ethical Statement
Ethics committee approval was not required for this study because of there was no study on animals or humans.
References
- Asselin, R. (1972). Frequency Filter for Time Integrations. Monthly Weather Review, 100(6), 487-490. https://doi.org/10.1175/15200493(1972)100<0487:FFFTI>2.3.CO;2
- Butcher, J. C. (2016). Numerical methods for ordinary differential equations (3. bs.). John Wiley and Sons, Ltd. https://doi.org/10.1002/9781119121534
- DeCaria, V., Gottlieb, S., Grant, Z. J., & Layton, W. J. (2022). A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD. Journal of Computational Physics, 455, Article 110927. https://doi.org/10.1016/j.jcp.2021.110927
- Durran, D. R. (2010). Numerical methods for fluid dynamics (2. bs., Cilt 32). Springer. https://doi.org/10.1007/978-1-4419-6412-0
- Griffiths, D. F., & Higham, D. J. (2010). Numerical methods for ordinary differential equations. Springer-Verlag London, Ltd. https://doi.org/10.1007/978-0-85729-148-6
- Guzel, A., & Layton, W. (2018). Time filters increase accuracy of the fully implicit method. BIT Numerical Mathematics, 58(2), 301–315. https://doi.org/10.1007/s10543-018-0695-z
- Guzel, A., & Trenchea, C. (2018). The Williams step increases the stability and accuracy of the hoRA time filter. Applied Numerical Mathematics, 131, 158–173. https://doi.org/10.1016/j.apnum.2018.05.003
- Hairer, E., Nørsett, S. P., & Wanner, G. (1993). Solving Ordinary Differential Equations I. In Springer Series in Computational Mathematics. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-78862-1
- Hurl, N., Layton, W., Li, Y., & Trenchea, C. (2014). Stability analysis of the Crank–Nicolson-Leapfrog method with the Robert–Asselin–Williams time filter. BIT Numerical Mathematics, 54(4), 1009–1021. https://doi.org/10.1007/s10543-014-0493-1
- Jury, E. (1964). Theory and Application of the Z-transform Method. John Wiley and Sons.
- Li, Y., & Trenchea, C. (2014). A higher-order Robert–Asselin type time filter. Journal of Computational Physics, 259, 23–32. https://doi.org/10.1016/j.jcp.2013.11.022
- McGovern, S. M. (2025). Adaptive step selection for a filtered implicit method. Journal of Scientific Computing, 103(2). https://doi.org/10.1007/s10915-025-02861-w
- Robert, A. J. (1966). The integration of a low order spectral form of the primitive meteorological equations. Journal of the Meteorological Society of Japan. Ser. II, 44(5), 237–245. https://doi.org/10.2151/jmsj1965.44.5_237
- Williams, P. D. (2009). A proposed modification to the Robert–Asselin time filter. Monthly Weather Review, 137(8), 2538–2546. https://doi.org/10.1175/2009MWR2724.1
- Williams, P. D. (2011). The RAW filter: An improvement to the Robert–Asselin filter in semi-implicit integrations. Monthly Weather Review, 139(6), 1996–2007. https://doi.org/10.1175/2010MWR3601.1
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.
Ethical Statement
Ethics committee approval was not required for this study because of there was no study on animals or humans.
References
- Asselin, R. (1972). Frequency Filter for Time Integrations. Monthly Weather Review, 100(6), 487-490. https://doi.org/10.1175/15200493(1972)100<0487:FFFTI>2.3.CO;2
- Butcher, J. C. (2016). Numerical methods for ordinary differential equations (3. bs.). John Wiley and Sons, Ltd. https://doi.org/10.1002/9781119121534
- DeCaria, V., Gottlieb, S., Grant, Z. J., & Layton, W. J. (2022). A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD. Journal of Computational Physics, 455, Article 110927. https://doi.org/10.1016/j.jcp.2021.110927
- Durran, D. R. (2010). Numerical methods for fluid dynamics (2. bs., Cilt 32). Springer. https://doi.org/10.1007/978-1-4419-6412-0
- Griffiths, D. F., & Higham, D. J. (2010). Numerical methods for ordinary differential equations. Springer-Verlag London, Ltd. https://doi.org/10.1007/978-0-85729-148-6
- Guzel, A., & Layton, W. (2018). Time filters increase accuracy of the fully implicit method. BIT Numerical Mathematics, 58(2), 301–315. https://doi.org/10.1007/s10543-018-0695-z
- Guzel, A., & Trenchea, C. (2018). The Williams step increases the stability and accuracy of the hoRA time filter. Applied Numerical Mathematics, 131, 158–173. https://doi.org/10.1016/j.apnum.2018.05.003
- Hairer, E., Nørsett, S. P., & Wanner, G. (1993). Solving Ordinary Differential Equations I. In Springer Series in Computational Mathematics. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-78862-1
- Hurl, N., Layton, W., Li, Y., & Trenchea, C. (2014). Stability analysis of the Crank–Nicolson-Leapfrog method with the Robert–Asselin–Williams time filter. BIT Numerical Mathematics, 54(4), 1009–1021. https://doi.org/10.1007/s10543-014-0493-1
- Jury, E. (1964). Theory and Application of the Z-transform Method. John Wiley and Sons.
- Li, Y., & Trenchea, C. (2014). A higher-order Robert–Asselin type time filter. Journal of Computational Physics, 259, 23–32. https://doi.org/10.1016/j.jcp.2013.11.022
- McGovern, S. M. (2025). Adaptive step selection for a filtered implicit method. Journal of Scientific Computing, 103(2). https://doi.org/10.1007/s10915-025-02861-w
- Robert, A. J. (1966). The integration of a low order spectral form of the primitive meteorological equations. Journal of the Meteorological Society of Japan. Ser. II, 44(5), 237–245. https://doi.org/10.2151/jmsj1965.44.5_237
- Williams, P. D. (2009). A proposed modification to the Robert–Asselin time filter. Monthly Weather Review, 137(8), 2538–2546. https://doi.org/10.1175/2009MWR2724.1
- Williams, P. D. (2011). The RAW filter: An improvement to the Robert–Asselin filter in semi-implicit integrations. Monthly Weather Review, 139(6), 1996–2007. https://doi.org/10.1175/2010MWR3601.1
Details
| Primary Language | English |
|---|---|
| Subjects | Ordinary Differential Equations, Difference Equations and Dynamical Systems |
| Journal Section | Research Article |
| Authors | |
| Submission Date | January 23, 2026 |
| Acceptance Date | February 25, 2026 |
| Publication Date | March 15, 2026 |
| DOI | https://doi.org/10.34248/bsengineering.1870475 |
| IZ | https://izlik.org/JA88DK26EJ |
| Published in Issue | Year 2026 Volume: 9 Issue: 2 |