A Fast Sine Transform Accelerated High-Order Finite Difference Method for Parabolic Problems over Irregular Domains

In this paper, a new Cartesian grid finite difference scheme is introduced for solving parabolic initial-boundary value problems involving irregular domains and Robin boundary condition in two and three dimensions. In spatial discretization, a ray-casting matched interface and boundary (MIB) method...

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Vydáno v:	Journal of scientific computing Ročník 95; číslo 2; s. 49
Hlavní autoři:	Li, Chuan, Ren, Yiming, Long, Guangqing, Boerman, Eric, Zhao, Shan
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.05.2023 Springer Nature B.V
Témata:	Algorithms Boundary conditions Boundary value problems Cartesian coordinates Computational Mathematics and Numerical Analysis Dirichlet problem Discretization Finite difference method Interfaces Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Numerical analysis Partial differential equations Theoretical Robin boundary condition Parabolic initial-boundary value problems 65M06 Fast Fourier transform (FFT) 65M85 Irregular domains 65M12 Matched interface and boundary (MIB)
ISSN:	0885-7474, 1573-7691
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Abstract	In this paper, a new Cartesian grid finite difference scheme is introduced for solving parabolic initial-boundary value problems involving irregular domains and Robin boundary condition in two and three dimensions. In spatial discretization, a ray-casting matched interface and boundary (MIB) method is utilized to enforce different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mixed combinations, along the normal direction to generate necessary fictitious values outside the irregular domain. This allows accurate approximations of jumps in derivatives at various boundary locations so that the fourth-order central difference can be corrected at all Cartesian nodes. By treating such corrections as additional unknowns, the order of finite difference discretization of the Laplacian operator can be preserved. Moreover, by constructing corrections for different types of irregular and corner points, the proposed augmented MIB (AMIB) method can accommodate complicated geometries, while maintaining the fourth order of accuracy in space. In temporal discretization, the standard Crank–Nicolson scheme is employed, which is second-order in time and unconditionally stable. Furthermore, a Fast Sine Transform acceleration algorithm is employed to efficiently invert the discrete Laplacian, so that the augmented linear system in each time step can be solved with a complexity of O ( N log N ) , where N stands for the total spatial degree-of-freedom. The accuracy, stability and efficiency of the proposed AMIB method are numerically validated by considering various parabolic problems in two and three dimensions.
AbstractList	In this paper, a new Cartesian grid finite difference scheme is introduced for solving parabolic initial-boundary value problems involving irregular domains and Robin boundary condition in two and three dimensions. In spatial discretization, a ray-casting matched interface and boundary (MIB) method is utilized to enforce different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mixed combinations, along the normal direction to generate necessary fictitious values outside the irregular domain. This allows accurate approximations of jumps in derivatives at various boundary locations so that the fourth-order central difference can be corrected at all Cartesian nodes. By treating such corrections as additional unknowns, the order of finite difference discretization of the Laplacian operator can be preserved. Moreover, by constructing corrections for different types of irregular and corner points, the proposed augmented MIB (AMIB) method can accommodate complicated geometries, while maintaining the fourth order of accuracy in space. In temporal discretization, the standard Crank–Nicolson scheme is employed, which is second-order in time and unconditionally stable. Furthermore, a Fast Sine Transform acceleration algorithm is employed to efficiently invert the discrete Laplacian, so that the augmented linear system in each time step can be solved with a complexity of O ( N log N ) , where N stands for the total spatial degree-of-freedom. The accuracy, stability and efficiency of the proposed AMIB method are numerically validated by considering various parabolic problems in two and three dimensions. In this paper, a new Cartesian grid finite difference scheme is introduced for solving parabolic initial-boundary value problems involving irregular domains and Robin boundary condition in two and three dimensions. In spatial discretization, a ray-casting matched interface and boundary (MIB) method is utilized to enforce different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mixed combinations, along the normal direction to generate necessary fictitious values outside the irregular domain. This allows accurate approximations of jumps in derivatives at various boundary locations so that the fourth-order central difference can be corrected at all Cartesian nodes. By treating such corrections as additional unknowns, the order of finite difference discretization of the Laplacian operator can be preserved. Moreover, by constructing corrections for different types of irregular and corner points, the proposed augmented MIB (AMIB) method can accommodate complicated geometries, while maintaining the fourth order of accuracy in space. In temporal discretization, the standard Crank–Nicolson scheme is employed, which is second-order in time and unconditionally stable. Furthermore, a Fast Sine Transform acceleration algorithm is employed to efficiently invert the discrete Laplacian, so that the augmented linear system in each time step can be solved with a complexity of O(NlogN), where N stands for the total spatial degree-of-freedom. The accuracy, stability and efficiency of the proposed AMIB method are numerically validated by considering various parabolic problems in two and three dimensions.
ArticleNumber	49
Author	Boerman, Eric Long, Guangqing Ren, Yiming Li, Chuan Zhao, Shan
Author_xml	– sequence: 1 givenname: Chuan surname: Li fullname: Li, Chuan organization: Department of Mathematics, West Chester University of Pennsylvania – sequence: 2 givenname: Yiming surname: Ren fullname: Ren, Yiming organization: Department of Mathematics, University of Alabama – sequence: 3 givenname: Guangqing surname: Long fullname: Long, Guangqing organization: Guangxi Key Lab of Human-Machine Interaction and Intelligent Decision, Nanning Normal University – sequence: 4 givenname: Eric surname: Boerman fullname: Boerman, Eric organization: Department of Mathematics, West Chester University of Pennsylvania – sequence: 5 givenname: Shan orcidid: 0000-0002-3023-2107 surname: Zhao fullname: Zhao, Shan email: szhao@ua.edu organization: Department of Mathematics, University of Alabama
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Snippet	In this paper, a new Cartesian grid finite difference scheme is introduced for solving parabolic initial-boundary value problems involving irregular domains...
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SubjectTerms	Algorithms Boundary conditions Boundary value problems Cartesian coordinates Computational Mathematics and Numerical Analysis Dirichlet problem Discretization Finite difference method Interfaces Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Numerical analysis Partial differential equations Theoretical
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Title	A Fast Sine Transform Accelerated High-Order Finite Difference Method for Parabolic Problems over Irregular Domains
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