Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications
This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagon...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Du, Qiang [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017transfer abstract |
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| Schlagwörter: |
Nonlocal Cahn–Hilliard equation |
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| Umfang: |
17 |
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| Übergeordnetes Werk: |
Enthalten in: Future-oriented repetitive thought, depressive symptoms, and suicide ideation severity: Role of future-event fluency and depressive predictive certainty - Miranda, Regina ELSEVIER, 2023, Amsterdam |
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| Übergeordnetes Werk: |
volume:332 ; year:2017 ; day:1 ; month:03 ; pages:118-134 ; extent:17 |
| Links: |
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| DOI / URN: |
10.1016/j.jcp.2016.11.028 |
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ELV030793327 |
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| 245 | 1 | 0 | |a Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications |
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| 520 | |a This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. | ||
| 520 | |a This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. | ||
| 650 | 7 | |a Nonlocal Cahn–Hilliard equation |2 Elsevier | |
| 650 | 7 | |a Nonlocal diffusion operator |2 Elsevier | |
| 650 | 7 | |a Nonlocal phase-field crystal model |2 Elsevier | |
| 650 | 7 | |a Peridynamic operator |2 Elsevier | |
| 650 | 7 | |a Fourier spectral method |2 Elsevier | |
| 650 | 7 | |a Nonlocal Allen–Cahn equation |2 Elsevier | |
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10.1016/j.jcp.2016.11.028 doi GBV00000000000100A.pica (DE-627)ELV030793327 (ELSEVIER)S0021-9991(16)30614-3 DE-627 ger DE-627 rakwb eng 530 510 000 530 DE-600 510 DE-600 000 DE-600 610 VZ 44.91 bkl Du, Qiang verfasserin aut Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications 2017transfer abstract 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. Nonlocal Cahn–Hilliard equation Elsevier Nonlocal diffusion operator Elsevier Nonlocal phase-field crystal model Elsevier Peridynamic operator Elsevier Fourier spectral method Elsevier Nonlocal Allen–Cahn equation Elsevier Yang, Jiang oth Enthalten in Elsevier Miranda, Regina ELSEVIER Future-oriented repetitive thought, depressive symptoms, and suicide ideation severity: Role of future-event fluency and depressive predictive certainty 2023 Amsterdam (DE-627)ELV010178430 volume:332 year:2017 day:1 month:03 pages:118-134 extent:17 https://doi.org/10.1016/j.jcp.2016.11.028 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_24 GBV_ILN_90 44.91 Psychiatrie Psychopathologie VZ AR 332 2017 1 0301 118-134 17 045F 530 |
| spelling |
10.1016/j.jcp.2016.11.028 doi GBV00000000000100A.pica (DE-627)ELV030793327 (ELSEVIER)S0021-9991(16)30614-3 DE-627 ger DE-627 rakwb eng 530 510 000 530 DE-600 510 DE-600 000 DE-600 610 VZ 44.91 bkl Du, Qiang verfasserin aut Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications 2017transfer abstract 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. Nonlocal Cahn–Hilliard equation Elsevier Nonlocal diffusion operator Elsevier Nonlocal phase-field crystal model Elsevier Peridynamic operator Elsevier Fourier spectral method Elsevier Nonlocal Allen–Cahn equation Elsevier Yang, Jiang oth Enthalten in Elsevier Miranda, Regina ELSEVIER Future-oriented repetitive thought, depressive symptoms, and suicide ideation severity: Role of future-event fluency and depressive predictive certainty 2023 Amsterdam (DE-627)ELV010178430 volume:332 year:2017 day:1 month:03 pages:118-134 extent:17 https://doi.org/10.1016/j.jcp.2016.11.028 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_24 GBV_ILN_90 44.91 Psychiatrie Psychopathologie VZ AR 332 2017 1 0301 118-134 17 045F 530 |
| allfields_unstemmed |
10.1016/j.jcp.2016.11.028 doi GBV00000000000100A.pica (DE-627)ELV030793327 (ELSEVIER)S0021-9991(16)30614-3 DE-627 ger DE-627 rakwb eng 530 510 000 530 DE-600 510 DE-600 000 DE-600 610 VZ 44.91 bkl Du, Qiang verfasserin aut Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications 2017transfer abstract 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. Nonlocal Cahn–Hilliard equation Elsevier Nonlocal diffusion operator Elsevier Nonlocal phase-field crystal model Elsevier Peridynamic operator Elsevier Fourier spectral method Elsevier Nonlocal Allen–Cahn equation Elsevier Yang, Jiang oth Enthalten in Elsevier Miranda, Regina ELSEVIER Future-oriented repetitive thought, depressive symptoms, and suicide ideation severity: Role of future-event fluency and depressive predictive certainty 2023 Amsterdam (DE-627)ELV010178430 volume:332 year:2017 day:1 month:03 pages:118-134 extent:17 https://doi.org/10.1016/j.jcp.2016.11.028 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_24 GBV_ILN_90 44.91 Psychiatrie Psychopathologie VZ AR 332 2017 1 0301 118-134 17 045F 530 |
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10.1016/j.jcp.2016.11.028 doi GBV00000000000100A.pica (DE-627)ELV030793327 (ELSEVIER)S0021-9991(16)30614-3 DE-627 ger DE-627 rakwb eng 530 510 000 530 DE-600 510 DE-600 000 DE-600 610 VZ 44.91 bkl Du, Qiang verfasserin aut Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications 2017transfer abstract 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. Nonlocal Cahn–Hilliard equation Elsevier Nonlocal diffusion operator Elsevier Nonlocal phase-field crystal model Elsevier Peridynamic operator Elsevier Fourier spectral method Elsevier Nonlocal Allen–Cahn equation Elsevier Yang, Jiang oth Enthalten in Elsevier Miranda, Regina ELSEVIER Future-oriented repetitive thought, depressive symptoms, and suicide ideation severity: Role of future-event fluency and depressive predictive certainty 2023 Amsterdam (DE-627)ELV010178430 volume:332 year:2017 day:1 month:03 pages:118-134 extent:17 https://doi.org/10.1016/j.jcp.2016.11.028 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_24 GBV_ILN_90 44.91 Psychiatrie Psychopathologie VZ AR 332 2017 1 0301 118-134 17 045F 530 |
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10.1016/j.jcp.2016.11.028 doi GBV00000000000100A.pica (DE-627)ELV030793327 (ELSEVIER)S0021-9991(16)30614-3 DE-627 ger DE-627 rakwb eng 530 510 000 530 DE-600 510 DE-600 000 DE-600 610 VZ 44.91 bkl Du, Qiang verfasserin aut Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications 2017transfer abstract 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. Nonlocal Cahn–Hilliard equation Elsevier Nonlocal diffusion operator Elsevier Nonlocal phase-field crystal model Elsevier Peridynamic operator Elsevier Fourier spectral method Elsevier Nonlocal Allen–Cahn equation Elsevier Yang, Jiang oth Enthalten in Elsevier Miranda, Regina ELSEVIER Future-oriented repetitive thought, depressive symptoms, and suicide ideation severity: Role of future-event fluency and depressive predictive certainty 2023 Amsterdam (DE-627)ELV010178430 volume:332 year:2017 day:1 month:03 pages:118-134 extent:17 https://doi.org/10.1016/j.jcp.2016.11.028 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_24 GBV_ILN_90 44.91 Psychiatrie Psychopathologie VZ AR 332 2017 1 0301 118-134 17 045F 530 |
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Enthalten in Future-oriented repetitive thought, depressive symptoms, and suicide ideation severity: Role of future-event fluency and depressive predictive certainty Amsterdam volume:332 year:2017 day:1 month:03 pages:118-134 extent:17 |
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Enthalten in Future-oriented repetitive thought, depressive symptoms, and suicide ideation severity: Role of future-event fluency and depressive predictive certainty Amsterdam volume:332 year:2017 day:1 month:03 pages:118-134 extent:17 |
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fast and accurate implementation of fourier spectral approximations of nonlocal diffusion operators and its applications |
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Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications |
| abstract |
This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. |
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This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. |
| abstract_unstemmed |
This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models. |
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Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications |
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