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Numerical Approaches for Advection-D...

Bi, Ran.

Numerical Approaches for Advection-Diffusion Equations Without and with an Interface.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Numerical Approaches for Advection-Diffusion Equations Without and with an Interface.
Author:	Bi, Ran.
Published:	Ann Arbor : ProQuest Dissertations & Theses, 2020
Description:	114 p.
Notes:	Source: Dissertations Abstracts International, Volume: 82-07, Section: B.
Notes:	Advisor: Liu, Yunan;Tran, Hien;Medhin, Negash;Li, Zhilin.
Contained By:	Dissertations Abstracts International82-07B.
Subject:	Operations research.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28122840
ISBN:	9798664736267

Numerical Approaches for Advection-Diffusion Equations Without and with an Interface.
Bi, Ran.

Numerical Approaches for Advection-Diffusion Equations Without and with an Interface. - Ann Arbor : ProQuest Dissertations & Theses, 2020 - 114 p.

Source: Dissertations Abstracts International, Volume: 82-07, Section: B.

Thesis (Ph.D.)--North Carolina State University, 2020.

This item must not be sold to any third party vendors.

Advection-diffusion partial differential equations with discontinuous and variable coefficients are widely used in many applications, such as financial mathematics and physics. Traditional numerical approaches can lead to some issues when the advection term dominates. This dissertation proposes two new finite difference methods for advection-diffusion models without and with interfaces under the domination of advection. The first method can be used to solve stationary and non-stationary advection-diffusion equations, while the second method can only be used for non-stationary cases.One method is to construct an integrating factor to transfer the original differential equation to a self-adjoint form then apply finite difference methods. Numerical integration and scaling may be needed to compute the integrating factor and solve the equation numerically. The newly developed finite difference method maintains second-order accuracy in both time and space with no nonphysical oscillations. The integrating factor approach is then applied to solve some interface problems using the immersed interface method (IIM). Elliptic and parabolic interface problems with piecewise constant coefficients and discontinuous variable coefficients that have a finite jump across a smooth interface are explained in detail. Second-order convergence proofs are accomplished for situations without and with an interface.The second approach uses the modified method of characteristic (MMOC) to eliminate the advection terms. This method has smaller time truncation errors and is computationally more efficient since it manifests almost hyperbolic nature of advection-dominated problems. The computed solution is second-order accurate in time and space. Extensions to problems with an interface are solved with a methodology as a combination of MMOC and IIM. Various interpolations and finite difference schemes are estimated. We believe that these two numerical approaches can be successfully employed in many other advection-diffusion models without and with an interface.

ISBN: 9798664736267Subjects--Topical Terms:

182516
Operations research.
Subjects--Index Terms:

Advection-diffusion partial differentialequation

Numerical Approaches for Advection-Diffusion Equations Without and with an Interface.
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245 1 0 $a Numerical Approaches for Advection-Diffusion Equations Without and with an Interface.
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300 $a 114 p.
500 $a Source: Dissertations Abstracts International, Volume: 82-07, Section: B.
500 $a Advisor: Liu, Yunan;Tran, Hien;Medhin, Negash;Li, Zhilin.
502 $a Thesis (Ph.D.)--North Carolina State University, 2020.
506 $a This item must not be sold to any third party vendors.
520 $a Advection-diffusion partial differential equations with discontinuous and variable coefficients are widely used in many applications, such as financial mathematics and physics. Traditional numerical approaches can lead to some issues when the advection term dominates. This dissertation proposes two new finite difference methods for advection-diffusion models without and with interfaces under the domination of advection. The first method can be used to solve stationary and non-stationary advection-diffusion equations, while the second method can only be used for non-stationary cases.One method is to construct an integrating factor to transfer the original differential equation to a self-adjoint form then apply finite difference methods. Numerical integration and scaling may be needed to compute the integrating factor and solve the equation numerically. The newly developed finite difference method maintains second-order accuracy in both time and space with no nonphysical oscillations. The integrating factor approach is then applied to solve some interface problems using the immersed interface method (IIM). Elliptic and parabolic interface problems with piecewise constant coefficients and discontinuous variable coefficients that have a finite jump across a smooth interface are explained in detail. Second-order convergence proofs are accomplished for situations without and with an interface.The second approach uses the modified method of characteristic (MMOC) to eliminate the advection terms. This method has smaller time truncation errors and is computationally more efficient since it manifests almost hyperbolic nature of advection-dominated problems. The computed solution is second-order accurate in time and space. Extensions to problems with an interface are solved with a methodology as a combination of MMOC and IIM. Various interpolations and finite difference schemes are estimated. We believe that these two numerical approaches can be successfully employed in many other advection-diffusion models without and with an interface.
590 $a School code: 0155.
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653 $a Advection-diffusion partial differentialequation
653 $a Immersed interface method
653 $a Modified method of characteristic
653 $a Finite difference schemes
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