Very high-order finite difference method on arbitrary geometries with Cartesian grids for non-linear convection diffusion reaction equations

• Published in: Journal of computational physics Vol. 498; p. 112667
• Main Authors: Clain, Stéphane, Lopes, Diogo, Pereira, Rui M. S., Pereira, Paulo A.
• Format: Journal Article
• Language: English
• Published: Elsevier 01-02-2024
• Subjects: Arbitrary geometries; Ciências Naturais; Finite difference; Immersed boundary; Matemáticas; ROD polynomial
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2023.112667

An arbitrary order finite difference method for solving non-linear convection Diffusion Reaction equations in curved boundary domains with Cartesian grid is proposed. Ghost points' values are determined with the Reconstruction Off-Site Data based on a polynomial interpolation using the least square method with constraints to enforce the boundary conditions. We propose a second-, fourth-, and sixth-order schemes for linear non-constant coefficients problem in both the conservative and non-conservative scalar equations. Extensions to non-linear scalar problems and systems are then implemented while preserving the optimal orders. Numerical simulations are carried out to provide evidence about the convergence order and the stability of the method.
R.M.S. Pereira acknowledges the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operational Fatores de Competitividade, and the National Funds through FCT – Fundação para a Ciência e a Tecnologia, project no. UID/FIS/04650/2019. P. A. Pereira acknowledges the financial support by Portuguese Funds through the Foundation for Science and Technology (FCT) in the framework of the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM. S. Clain acknowledges the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operational Fatores de Competitividade, and the National Funds through FCT – Fundação para a Ciência e a Tecnologia, project Nº UIDB/00324/2020.