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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

DOI: 10.4208/nmtma.OA-2019-0143

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, convection-diffusion equation, local truncation error.

  • AMS Subject Headings

Primary Classification: 65N06, Secondary Classification: 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-13-881, author = {Esmaeilzadeh , M.Barron , R.M. and Balachandar , R.}, 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 - Esmaeilzadeh , M. AU - Barron , R.M. AU - Balachandar , R. 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, convection-diffusion 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.

Esmaeilzadeh , M.Barron , R.M. and Balachandar , R.. (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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