### Abstract

The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.

Original language | English (US) |
---|---|

Pages (from-to) | 4996-5010 |

Number of pages | 15 |

Journal | Journal of Computational Physics |

Volume | 229 |

Issue number | 13 |

DOIs | |

State | Published - Jul 2010 |

Externally published | Yes |

### Fingerprint

### Keywords

- Diffusion process
- Finite element methods
- Krylov deferred correction methods
- Multiscale modeling
- Semi-implicit preconditioner

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy (miscellaneous)

### Cite this

*Journal of Computational Physics*,

*229*(13), 4996-5010. https://doi.org/10.1016/j.jcp.2010.03.021

**An evaluation of solution algorithms and numerical approximation methods for modeling an ion exchange process.** / Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor; Miller, Cass T.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 229, no. 13, pp. 4996-5010. https://doi.org/10.1016/j.jcp.2010.03.021

}

TY - JOUR

T1 - An evaluation of solution algorithms and numerical approximation methods for modeling an ion exchange process

AU - Bu, Sunyoung

AU - Huang, Jingfang

AU - Boyer, Treavor

AU - Miller, Cass T.

PY - 2010/7

Y1 - 2010/7

N2 - The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.

AB - The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.

KW - Diffusion process

KW - Finite element methods

KW - Krylov deferred correction methods

KW - Multiscale modeling

KW - Semi-implicit preconditioner

UR - http://www.scopus.com/inward/record.url?scp=77952421449&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952421449&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2010.03.021

DO - 10.1016/j.jcp.2010.03.021

M3 - Article

AN - SCOPUS:77952421449

VL - 229

SP - 4996

EP - 5010

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 13

ER -