Volume 13, Issue 4
Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part I: Concepts and Fundamentals

M. Esmaeilzadeh, R.M. Barron & R. Balachandar

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 881-907.

Published online: 2020-06

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  • Abstract

A new finite difference (FD) method, referred to as "Cartesian cut-stencil FD", is introduced to obtain the numerical solution of partial differential equations on any arbitrary irregular shaped domain. The 2nd-order accurate two-dimensional Cartesian cut-stencil FD method utilizes a 5-point stencil and relies on the construction of a unique mapping of each physical stencil, rather than a cell, in any arbitrary domain to a generic uniform computational stencil. The treatment of boundary conditions and quantification of the solution accuracy using the local truncation error are discussed. Numerical solutions of the steady convection-diffusion equation on sample complex domains have been obtained and the results have been compared to exact solutions for manufactured partial differential equations (PDEs) and other numerical solutions.

  • Keywords

Cartesian cut-stencils, finite difference method, irregular domains, convectiondiffusion equation, local truncation error.

  • AMS Subject Headings

Primary Classification: 65N06, Secondary Classification: 35Q35

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-13-881, author = {M. Esmaeilzadeh , and R.M. Barron , and R. Balachandar , }, title = {Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part I: Concepts and Fundamentals}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {4}, pages = {881--907}, abstract = {

A new finite difference (FD) method, referred to as "Cartesian cut-stencil FD", is introduced to obtain the numerical solution of partial differential equations on any arbitrary irregular shaped domain. The 2nd-order accurate two-dimensional Cartesian cut-stencil FD method utilizes a 5-point stencil and relies on the construction of a unique mapping of each physical stencil, rather than a cell, in any arbitrary domain to a generic uniform computational stencil. The treatment of boundary conditions and quantification of the solution accuracy using the local truncation error are discussed. Numerical solutions of the steady convection-diffusion equation on sample complex domains have been obtained and the results have been compared to exact solutions for manufactured partial differential equations (PDEs) and other numerical solutions.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0143}, url = {http://global-sci.org/intro/article_detail/nmtma/16958.html} }
TY - JOUR T1 - Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part I: Concepts and Fundamentals AU - M. Esmaeilzadeh , AU - R.M. Barron , AU - R. Balachandar , JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 881 EP - 907 PY - 2020 DA - 2020/06 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0143 UR - https://global-sci.org/intro/article_detail/nmtma/16958.html KW - Cartesian cut-stencils, finite difference method, irregular domains, convectiondiffusion equation, local truncation error. AB -

A new finite difference (FD) method, referred to as "Cartesian cut-stencil FD", is introduced to obtain the numerical solution of partial differential equations on any arbitrary irregular shaped domain. The 2nd-order accurate two-dimensional Cartesian cut-stencil FD method utilizes a 5-point stencil and relies on the construction of a unique mapping of each physical stencil, rather than a cell, in any arbitrary domain to a generic uniform computational stencil. The treatment of boundary conditions and quantification of the solution accuracy using the local truncation error are discussed. Numerical solutions of the steady convection-diffusion equation on sample complex domains have been obtained and the results have been compared to exact solutions for manufactured partial differential equations (PDEs) and other numerical solutions.

M. Esmaeilzadeh, R.M. Barron & R. Balachandar. (2020). Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part I: Concepts and Fundamentals. Numerical Mathematics: Theory, Methods and Applications. 13 (4). 881-907. doi:10.4208/nmtma.OA-2019-0143
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