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**Simple, accurate, and efficient revisions to MacCormack and Saulyev schemes: high Péclet numbers.**
*(English)*
Zbl 1114.65103

Summary: Stream water quality modeling often involves numerical methods to solve the dynamic one-dimensional advection-dispersion-reaction equations (ADRE). There are numerous explicit and implicit finite difference schemes for solving these problems, and two commonly used schemes are the MacCormack and Saulyev schemes.

This paper presents simple revisions to these schemes that make them more accurate without significant loss of computation efficiency. Using advection dominated (high Péclet number) problems as test cases, performances of the revised schemes are compared to performances of five classic schemes: forward-time/centered-space (FTCS); backward-time/centered-space (BTCS); Crank-Nicolson; and the traditional MacCormack and Saulyev schemes. All seven of the above numerical schemes are tested against analytical solutions for pulse and step inputs of mass to a steady flow in a channel, and performances are considered with respect to stability, accuracy, and computational efficiency.

Results indicate that both the modified Saulyev and the MacCormack schemes, which are named the Saulyev\(_{c}\) and MacCormack\(_{c}\) schemes, respectively, greatly improve the prediction accuracy over the original ones. The computation efficiency in terms of CPU time was not impacted for the Saulyev\(_{c}\) scheme. The MacCormack\(_{c}\) scheme demonstrates increased time consumption but is still much faster than implicit schemes.

This paper presents simple revisions to these schemes that make them more accurate without significant loss of computation efficiency. Using advection dominated (high Péclet number) problems as test cases, performances of the revised schemes are compared to performances of five classic schemes: forward-time/centered-space (FTCS); backward-time/centered-space (BTCS); Crank-Nicolson; and the traditional MacCormack and Saulyev schemes. All seven of the above numerical schemes are tested against analytical solutions for pulse and step inputs of mass to a steady flow in a channel, and performances are considered with respect to stability, accuracy, and computational efficiency.

Results indicate that both the modified Saulyev and the MacCormack schemes, which are named the Saulyev\(_{c}\) and MacCormack\(_{c}\) schemes, respectively, greatly improve the prediction accuracy over the original ones. The computation efficiency in terms of CPU time was not impacted for the Saulyev\(_{c}\) scheme. The MacCormack\(_{c}\) scheme demonstrates increased time consumption but is still much faster than implicit schemes.

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

### Keywords:

finite differences; MacCormack scheme; Saulyev scheme; one-dimensional advection-dispersion-reaction equations; comparison of methods; forward-time/centered-space scheme; backward-time/centered-space scheme; Crank-Nicolson scheme; numerical example; stream water quality modeling
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\textit{G. Li} and \textit{C. R. Jackson}, Appl. Math. Comput. 186, No. 1, 610--622 (2007; Zbl 1114.65103)

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