Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm
The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one-...
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| Published in | Journal of computational physics Vol. 468; p. 111513 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge
Elsevier Inc
01.11.2022
Elsevier Science Ltd |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0021-9991 1090-2716 |
| DOI | 10.1016/j.jcp.2022.111513 |
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| Abstract | The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.
•A fast approximation algorithm for variable-order spectral fractional Laplacian.•Construction of solution landscapes of space-fractional phase field models.•Reconfiguration of fractional Laplacian problems by integer-order problems. |
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| AbstractList | The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems. The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems. •A fast approximation algorithm for variable-order spectral fractional Laplacian.•Construction of solution landscapes of space-fractional phase field models.•Reconfiguration of fractional Laplacian problems by integer-order problems. |
| ArticleNumber | 111513 |
| Author | Zheng, Xiangcheng Yu, Bing Zhang, Pingwen Zhang, Lei |
| Author_xml | – sequence: 1 givenname: Bing orcidid: 0000-0001-8494-3775 surname: Yu fullname: Yu, Bing email: yubing93@pku.edu.cn organization: School of Mathematical Sciences, Peking University, Beijing 100871, China – sequence: 2 givenname: Xiangcheng surname: Zheng fullname: Zheng, Xiangcheng email: zhengxch@math.pku.edu.cn organization: School of Mathematical Sciences, Peking University, Beijing 100871, China – sequence: 3 givenname: Pingwen surname: Zhang fullname: Zhang, Pingwen email: pzhang@pku.edu.cn organization: School of Mathematical Sciences, Laboratory of Mathematics and Applied Mathematics, Peking University, Beijing 100871, China – sequence: 4 givenname: Lei orcidid: 0000-0001-9972-2051 surname: Zhang fullname: Zhang, Lei email: zhangl@math.pku.edu.cn organization: Beijing International Center for Mathematical Research, Center for Machine Learning Research, Center for Quantitative Biology, Peking University, Beijing 100871, China |
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| Cites_doi | 10.1016/j.jcp.2019.109141 10.1088/1361-6544/abc5d4 10.1137/080730597 10.1051/m2an/2013091 10.1137/19M1243750 10.1137/19M1296720 10.1002/num.22865 10.1016/j.jconhyd.2013.11.002 10.1017/S0962492921000088 10.1137/16M1097109 10.1137/16M1075302 10.1016/j.jcp.2019.01.042 10.1017/S096249292000001X 10.1137/141001299 10.1137/140972676 10.1016/j.cma.2020.112936 10.1088/0951-7715/24/6/008 10.1137/15M1033952 10.1137/19M128377X 10.1137/19M1253356 10.1007/s10915-021-01427-w 10.1016/j.cma.2016.03.018 10.1146/annurev.matsci.32.112001.132041 10.1007/s11425-020-1737-1 10.1103/PhysRevLett.124.090601 10.1137/18M1224970 10.1007/s10915-020-01366-y 10.1016/j.jcp.2015.07.011 10.1103/PhysRevLett.98.265703 10.1137/140984798 10.1016/j.jcp.2013.06.031 10.1137/19M1245621 10.1063/1.1436470 10.1023/A:1016586905654 10.1016/j.physa.2009.07.024 10.1007/s00791-018-0289-y 10.1137/130933745 10.1137/19M1244299 10.1080/00036811.2020.1789596 10.1016/j.physd.2021.133081 10.1103/PhysRevE.84.025702 10.1098/rspa.2021.0458 10.1016/j.jde.2020.12.034 |
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| Keywords | Stationary solution Variable-order Saddle dynamics Fractional Laplacian Solution landscape Phase field |
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| References | Yin, Zhang, Zhang (br0420) 2019; 41 Doye, Wales (br0130) 2002; 116 Lanzara, Maz'ya, Schmidt (br0230) 2021 Sun, Chen, Chen (br0330) 2009; 388 D'Elia, Glusa (br0100) 2022 Wang, Zhang, Zhang (br0370) 2021; 30 Zhuang, Liu, Anh, Turner (br0520) 2009; 47 D'Elia, Du, Glusa, Gunzburger, Tian, Zhou (br0110) 2020; 29 Antil, Rautenberg (br0040) 2019; 51 Zayernouri, Karniadakis (br0440) 2013; 252 Ainsworth, Mao (br0060) 2017; 55 Yin, Yu, Zhang (br0410) 2021; 64 Sun, Zhang, Chen, Reeves (br0340) 2014; 157 Farrell, Birkisson, Funke (br0220) 2015; 37 Song, Xu (br0320) 2015; 299 Duo, Wang (br0140) 2019; 384 Mao, Shen (br0240) 2017; 39 Ervin (br0170) 2021; 278 Han, Yin, Hu, Majumdar, Zhang (br0190) 2021; 477 Sheng, Shen, Tang, Wang, Yuan (br0310) 2020; 58 Wiggins (br0380) 2003 Tian, Du (br0360) 2020; 62 Du, Ju, Li, Qiao (br0150) 2021; 63 Zheng, Wang (br0500) 2020; 58 Adams, Fournier (br0020) 2003 Deng, Hesthaven (br0120) 2013; 47 Yin, Wang, Chen, Zhang, Zhang (br0400) 2020; 124 Milnor (br0260) 1963 Mehta (br0250) 2011; 84 Xu, Darve (br0390) 2020; 364 Zheng, Ervin, Wang (br0510) 2021; 86 Pang, Sun (br0270) 2021; 87 Farquhar, Moroney, Yang, Turner, Burrage (br0210) Shen, Tang, Wang (br0300) 2011; vol. 41 Bonito, Borthagaray, Nochetto, Otárola, Salgado (br0070) 2018; 19 E, Zhou (br0160) 2011; 24 Qiao, Sun (br0280) 2014; 36 Zhang, Du, Zheng (br0470) 2016; 38 Tang, Wang, Yuan, Zhou (br0350) 2020; 42 Zeng, Zhang, Karniadakis (br0450) 2015; 37 Zhang, Jiang, Zeng, Karniadakis (br0480) 2020; 405 Allgower, Georg (br0030) 2003 Chen (br0080) 2002; 32 Han, Yin, Zhang, Majumdar, Zhang (br0180) 2021; 34 Song, Xu, Karniadakis (br0290) 2016; 305 Acosta, Borthagaray (br0010) 2017; 55 Yin, Zhang, Zhang (br0430) 2022; 430 Zhang, Chen, Du (br0460) 2007; 98 Lorenzo, Hartley (br0200) 2002; 29 Antil, Ceretani, Rautenberg (br0050) 2021 Zheng, Wang (br0490) 2022; 101 Colli, Gilardi (br0090) 2019; 114 Duo (10.1016/j.jcp.2022.111513_br0140) 2019; 384 Farquhar (10.1016/j.jcp.2022.111513_br0210) Mehta (10.1016/j.jcp.2022.111513_br0250) 2011; 84 Milnor (10.1016/j.jcp.2022.111513_br0260) 1963 Wang (10.1016/j.jcp.2022.111513_br0370) 2021; 30 Antil (10.1016/j.jcp.2022.111513_br0050) D'Elia (10.1016/j.jcp.2022.111513_br0100) 2022 Bonito (10.1016/j.jcp.2022.111513_br0070) 2018; 19 Du (10.1016/j.jcp.2022.111513_br0150) 2021; 63 Lorenzo (10.1016/j.jcp.2022.111513_br0200) 2002; 29 Lanzara (10.1016/j.jcp.2022.111513_br0230) 2021 Tang (10.1016/j.jcp.2022.111513_br0350) 2020; 42 Tian (10.1016/j.jcp.2022.111513_br0360) 2020; 62 Pang (10.1016/j.jcp.2022.111513_br0270) 2021; 87 Zheng (10.1016/j.jcp.2022.111513_br0490) 2022; 101 Sun (10.1016/j.jcp.2022.111513_br0340) 2014; 157 Wiggins (10.1016/j.jcp.2022.111513_br0380) 2003 Yin (10.1016/j.jcp.2022.111513_br0410) 2021; 64 Mao (10.1016/j.jcp.2022.111513_br0240) 2017; 39 Ervin (10.1016/j.jcp.2022.111513_br0170) 2021; 278 Xu (10.1016/j.jcp.2022.111513_br0390) 2020; 364 Song (10.1016/j.jcp.2022.111513_br0290) 2016; 305 Ainsworth (10.1016/j.jcp.2022.111513_br0060) 2017; 55 Colli (10.1016/j.jcp.2022.111513_br0090) 2019; 114 Zhang (10.1016/j.jcp.2022.111513_br0460) 2007; 98 Zhuang (10.1016/j.jcp.2022.111513_br0520) 2009; 47 Han (10.1016/j.jcp.2022.111513_br0180) 2021; 34 Sheng (10.1016/j.jcp.2022.111513_br0310) 2020; 58 Zhang (10.1016/j.jcp.2022.111513_br0480) 2020; 405 Yin (10.1016/j.jcp.2022.111513_br0420) 2019; 41 Chen (10.1016/j.jcp.2022.111513_br0080) 2002; 32 Deng (10.1016/j.jcp.2022.111513_br0120) 2013; 47 Shen (10.1016/j.jcp.2022.111513_br0300) 2011; vol. 41 Zayernouri (10.1016/j.jcp.2022.111513_br0440) 2013; 252 Han (10.1016/j.jcp.2022.111513_br0190) 2021; 477 Farrell (10.1016/j.jcp.2022.111513_br0220) 2015; 37 Zhang (10.1016/j.jcp.2022.111513_br0470) 2016; 38 Song (10.1016/j.jcp.2022.111513_br0320) 2015; 299 Sun (10.1016/j.jcp.2022.111513_br0330) 2009; 388 Yin (10.1016/j.jcp.2022.111513_br0400) 2020; 124 Adams (10.1016/j.jcp.2022.111513_br0020) 2003 D'Elia (10.1016/j.jcp.2022.111513_br0110) 2020; 29 E (10.1016/j.jcp.2022.111513_br0160) 2011; 24 Zeng (10.1016/j.jcp.2022.111513_br0450) 2015; 37 Acosta (10.1016/j.jcp.2022.111513_br0010) 2017; 55 Qiao (10.1016/j.jcp.2022.111513_br0280) 2014; 36 Allgower (10.1016/j.jcp.2022.111513_br0030) 2003 Zheng (10.1016/j.jcp.2022.111513_br0500) 2020; 58 Yin (10.1016/j.jcp.2022.111513_br0430) 2022; 430 Antil (10.1016/j.jcp.2022.111513_br0040) 2019; 51 Doye (10.1016/j.jcp.2022.111513_br0130) 2002; 116 Zheng (10.1016/j.jcp.2022.111513_br0510) 2021; 86 |
| References_xml | – volume: 37 start-page: A2026 year: 2015 end-page: A2045 ident: br0220 article-title: Deflation techniques for finding distinct solutions of nonlinear partial differential equations publication-title: SIAM J. Sci. Comput. – volume: 388 start-page: 4586 year: 2009 end-page: 4592 ident: br0330 article-title: Variable-order fractional differential operators in anomalous diffusion modeling publication-title: Phys. A, Stat. Mech. Appl. – volume: 55 start-page: 1689 year: 2017 end-page: 1718 ident: br0060 article-title: Analysis and approximation of a fractional Cahn-Hilliard equation publication-title: SIAM J. Numer. Anal. – volume: 32 start-page: 113 year: 2002 end-page: 140 ident: br0080 article-title: Phase-field models for microstructure evolution publication-title: Annu. Rev. Mater. Res. – volume: 58 start-page: 2435 year: 2020 end-page: 2464 ident: br0310 article-title: Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains publication-title: SIAM J. Numer. Anal. – volume: 39 start-page: A1928 year: 2017 end-page: A1950 ident: br0240 article-title: Hermite spectral methods for fractional PDEs in unbounded domains publication-title: SIAM J. Sci. Comput. – volume: 30 start-page: 765 year: 2021 end-page: 851 ident: br0370 article-title: Modeling and computation of liquid crystals publication-title: Acta Numer. – volume: 384 start-page: 253 year: 2019 end-page: 269 ident: br0140 article-title: A fractional phase-field model using an infinitesimal generator of publication-title: J. Comput. Phys. – volume: 101 start-page: 1848 year: 2022 end-page: 1870 ident: br0490 article-title: Variable-order space-fractional diffusion equations and a variable-order modification of constant-order fractional problems publication-title: Appl. Anal. – volume: 62 start-page: 199 year: 2020 end-page: 227 ident: br0360 article-title: Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models publication-title: SIAM Rev. – volume: 41 start-page: A3576 year: 2019 end-page: A3595 ident: br0420 article-title: High-index optimization-based shrinking dimer method for finding high-index saddle points publication-title: SIAM J. Sci. Comput. – volume: 278 start-page: 294 year: 2021 end-page: 325 ident: br0170 article-title: Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces publication-title: J. Differ. Equ. – volume: vol. 41 year: 2011 ident: br0300 article-title: Spectral Methods: Algorithms, Analysis and Applications publication-title: Springer Ser. Comput. Math. – year: 2022 ident: br0100 article-title: A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems publication-title: Numer. Methods Partial Differ. Eq. – volume: 364 year: 2020 ident: br0390 article-title: Isogeometric collocation method for the fractional Laplacian in the 2D bounded domain publication-title: Comput. Methods Appl. Mech. Eng. – volume: 29 start-page: 1 year: 2020 end-page: 124 ident: br0110 article-title: Numerical methods for nonlocal and fractional models publication-title: Acta Numer. – volume: 124 year: 2020 ident: br0400 article-title: Construction of a pathway map on a complicated energy landscape publication-title: Phys. Rev. Lett. – volume: 116 start-page: 3777 year: 2002 end-page: 3788 ident: br0130 article-title: Saddle points and dynamics of Lennard-Jones clusters, solids, and supercooled liquids publication-title: J. Chem. Phys. – year: 2003 ident: br0380 article-title: Introduction to Applied Nonlinear Dynamical Systems and Chaos – volume: 51 start-page: 2479 year: 2019 end-page: 2503 ident: br0040 article-title: Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications publication-title: SIAM J. Math. Anal. – volume: 63 start-page: 317 year: 2021 end-page: 359 ident: br0150 article-title: Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes publication-title: SIAM Rev. – volume: 55 start-page: 472 year: 2017 end-page: 495 ident: br0010 article-title: A fractional Laplace equation: regularity of solutions and finite element approximations publication-title: SIAM J. Numer. Anal. – volume: 98 year: 2007 ident: br0460 article-title: Morphology of critical nuclei in solid-state phase transformations publication-title: Phys. Rev. Lett. – volume: 29 start-page: 57 year: 2002 end-page: 98 ident: br0200 article-title: Variable order and distributed order fractional operators publication-title: Nonlinear Dyn. – volume: 114 start-page: 93 year: 2019 end-page: 128 ident: br0090 article-title: Well-posedness, regularity and asymptotic analyses for a fractional phase field system publication-title: Asymptot. Anal. – volume: 47 start-page: 1845 year: 2013 end-page: 1864 ident: br0120 article-title: Local discontinuous Galerkin methods for fractional diffusion equations publication-title: ESAIM: M2AN – start-page: 1 year: 2021 end-page: 17 ident: br0230 article-title: Fast computation of the multidimensional fractional Laplacian publication-title: Appl. Anal. – volume: 36 start-page: 708 year: 2014 end-page: 728 ident: br0280 article-title: Two-phase fluid simulation using a diffuse interface model with Peng–Robinson equation of state publication-title: SIAM J. Sci. Comput. – ident: br0210 article-title: Computational modelling of cardiac ischaemia using a variable-order fractional Laplacian – volume: 42 start-page: A585 year: 2020 end-page: A611 ident: br0350 article-title: Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains publication-title: SIAM J. Sci. Comput. – volume: 58 start-page: 330 year: 2020 end-page: 352 ident: br0500 article-title: An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes publication-title: SIAM J. Numer. Anal. – volume: 299 start-page: 196 year: 2015 end-page: 214 ident: br0320 article-title: Spectral direction splitting methods for two-dimensional space fractional diffusion equations publication-title: J. Comput. Phys. – volume: 37 start-page: A2710 year: 2015 end-page: A2732 ident: br0450 article-title: A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations publication-title: SIAM J. Sci. Comput. – volume: 477 start-page: 20210458 year: 2021 ident: br0190 article-title: Solution landscapes of the simplified Ericksen-Leslie model and its comparison with the reduced Landau-de Gennes model publication-title: Proc. R. Soc. A – volume: 38 start-page: A528 year: 2016 end-page: A544 ident: br0470 article-title: Optimization-based shrinking dimer method for finding transition states publication-title: SIAM J. Sci. Comput. – volume: 34 start-page: 2048 year: 2021 end-page: 2069 ident: br0180 article-title: Solution landscape of a reduced Landau–de Gennes model on a hexagon publication-title: Nonlinearity – year: 2021 ident: br0050 article-title: The spatially variant fractional Laplacian – year: 2003 ident: br0030 article-title: Introduction to Numerical Continuation Methods – volume: 305 start-page: 376 year: 2016 end-page: 404 ident: br0290 article-title: A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations publication-title: Comput. Methods Appl. Mech. Eng. – volume: 84 year: 2011 ident: br0250 article-title: Finding all the stationary points of a potential-energy landscape via numerical polynomial-homotopy-continuation method publication-title: Phys. Rev. E – volume: 64 start-page: 1801 year: 2021 end-page: 1816 ident: br0410 article-title: Searching the solution landscape by generalized high-index saddle dynamics publication-title: Sci. China Math. – volume: 87 start-page: 15 year: 2021 ident: br0270 article-title: A fast algorithm for the variable-order spatial fractional advection-diffusion equation publication-title: J. Sci. Comput. – year: 2003 ident: br0020 article-title: Sobolev Spaces – volume: 47 start-page: 1760 year: 2009 end-page: 1781 ident: br0520 article-title: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term publication-title: SIAM J. Numer. Anal. – year: 1963 ident: br0260 article-title: Morse Theory – volume: 430 year: 2022 ident: br0430 article-title: Solution landscape of the Onsager model identifies non-axisymmetric critical points publication-title: Phys. D, Nonlin. Phenom. – volume: 86 start-page: 29 year: 2021 ident: br0510 article-title: Optimal Petrov-Galerkin spectral approximation method for the fractional diffusion, advection, reaction equation on a bounded interval publication-title: J. Sci. Comput. – volume: 157 start-page: 47 year: 2014 end-page: 58 ident: br0340 article-title: Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media publication-title: J. Contam. Hydrol. – volume: 24 start-page: 1831 year: 2011 end-page: 1842 ident: br0160 article-title: The gentlest ascent dynamics publication-title: Nonlinearity – volume: 252 start-page: 495 year: 2013 end-page: 517 ident: br0440 article-title: Fractional Sturm-Liouville eigen-problems: theory and numerical approximation publication-title: J. Comput. Phys. – volume: 405 year: 2020 ident: br0480 article-title: A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations publication-title: J. Comput. Phys. – volume: 19 start-page: 19 year: 2018 end-page: 46 ident: br0070 article-title: Numerical methods for fractional diffusion publication-title: Comput. Vis. Sci. – volume: 405 year: 2020 ident: 10.1016/j.jcp.2022.111513_br0480 article-title: A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.109141 – volume: 34 start-page: 2048 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0180 article-title: Solution landscape of a reduced Landau–de Gennes model on a hexagon publication-title: Nonlinearity doi: 10.1088/1361-6544/abc5d4 – volume: 47 start-page: 1760 year: 2009 ident: 10.1016/j.jcp.2022.111513_br0520 article-title: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term publication-title: SIAM J. Numer. Anal. doi: 10.1137/080730597 – volume: 47 start-page: 1845 year: 2013 ident: 10.1016/j.jcp.2022.111513_br0120 article-title: Local discontinuous Galerkin methods for fractional diffusion equations publication-title: ESAIM: M2AN doi: 10.1051/m2an/2013091 – volume: 63 start-page: 317 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0150 article-title: Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes publication-title: SIAM Rev. doi: 10.1137/19M1243750 – volume: 62 start-page: 199 year: 2020 ident: 10.1016/j.jcp.2022.111513_br0360 article-title: Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models publication-title: SIAM Rev. doi: 10.1137/19M1296720 – year: 2022 ident: 10.1016/j.jcp.2022.111513_br0100 article-title: A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems publication-title: Numer. Methods Partial Differ. Eq. doi: 10.1002/num.22865 – volume: 157 start-page: 47 year: 2014 ident: 10.1016/j.jcp.2022.111513_br0340 article-title: Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media publication-title: J. Contam. Hydrol. doi: 10.1016/j.jconhyd.2013.11.002 – volume: 30 start-page: 765 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0370 article-title: Modeling and computation of liquid crystals publication-title: Acta Numer. doi: 10.1017/S0962492921000088 – ident: 10.1016/j.jcp.2022.111513_br0050 – volume: 39 start-page: A1928 year: 2017 ident: 10.1016/j.jcp.2022.111513_br0240 article-title: Hermite spectral methods for fractional PDEs in unbounded domains publication-title: SIAM J. Sci. Comput. doi: 10.1137/16M1097109 – volume: vol. 41 year: 2011 ident: 10.1016/j.jcp.2022.111513_br0300 article-title: Spectral Methods: Algorithms, Analysis and Applications – volume: 55 start-page: 1689 year: 2017 ident: 10.1016/j.jcp.2022.111513_br0060 article-title: Analysis and approximation of a fractional Cahn-Hilliard equation publication-title: SIAM J. Numer. Anal. doi: 10.1137/16M1075302 – volume: 384 start-page: 253 year: 2019 ident: 10.1016/j.jcp.2022.111513_br0140 article-title: A fractional phase-field model using an infinitesimal generator of α stable Lévy process publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.01.042 – year: 2003 ident: 10.1016/j.jcp.2022.111513_br0030 – volume: 29 start-page: 1 year: 2020 ident: 10.1016/j.jcp.2022.111513_br0110 article-title: Numerical methods for nonlocal and fractional models publication-title: Acta Numer. doi: 10.1017/S096249292000001X – volume: 37 start-page: A2710 year: 2015 ident: 10.1016/j.jcp.2022.111513_br0450 article-title: A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations publication-title: SIAM J. Sci. Comput. doi: 10.1137/141001299 – volume: 38 start-page: A528 year: 2016 ident: 10.1016/j.jcp.2022.111513_br0470 article-title: Optimization-based shrinking dimer method for finding transition states publication-title: SIAM J. Sci. Comput. doi: 10.1137/140972676 – year: 2003 ident: 10.1016/j.jcp.2022.111513_br0020 – volume: 364 year: 2020 ident: 10.1016/j.jcp.2022.111513_br0390 article-title: Isogeometric collocation method for the fractional Laplacian in the 2D bounded domain publication-title: Comput. Methods Appl. Mech. Eng. doi: 10.1016/j.cma.2020.112936 – volume: 24 start-page: 1831 year: 2011 ident: 10.1016/j.jcp.2022.111513_br0160 article-title: The gentlest ascent dynamics publication-title: Nonlinearity doi: 10.1088/0951-7715/24/6/008 – volume: 55 start-page: 472 year: 2017 ident: 10.1016/j.jcp.2022.111513_br0010 article-title: A fractional Laplace equation: regularity of solutions and finite element approximations publication-title: SIAM J. Numer. Anal. doi: 10.1137/15M1033952 – volume: 58 start-page: 2435 year: 2020 ident: 10.1016/j.jcp.2022.111513_br0310 article-title: Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains publication-title: SIAM J. Numer. Anal. doi: 10.1137/19M128377X – volume: 41 start-page: A3576 year: 2019 ident: 10.1016/j.jcp.2022.111513_br0420 article-title: High-index optimization-based shrinking dimer method for finding high-index saddle points publication-title: SIAM J. Sci. Comput. doi: 10.1137/19M1253356 – year: 1963 ident: 10.1016/j.jcp.2022.111513_br0260 – volume: 87 start-page: 15 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0270 article-title: A fast algorithm for the variable-order spatial fractional advection-diffusion equation publication-title: J. Sci. Comput. doi: 10.1007/s10915-021-01427-w – volume: 305 start-page: 376 year: 2016 ident: 10.1016/j.jcp.2022.111513_br0290 article-title: A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations publication-title: Comput. Methods Appl. Mech. Eng. doi: 10.1016/j.cma.2016.03.018 – year: 2003 ident: 10.1016/j.jcp.2022.111513_br0380 – volume: 32 start-page: 113 year: 2002 ident: 10.1016/j.jcp.2022.111513_br0080 article-title: Phase-field models for microstructure evolution publication-title: Annu. Rev. Mater. Res. doi: 10.1146/annurev.matsci.32.112001.132041 – volume: 64 start-page: 1801 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0410 article-title: Searching the solution landscape by generalized high-index saddle dynamics publication-title: Sci. China Math. doi: 10.1007/s11425-020-1737-1 – volume: 124 year: 2020 ident: 10.1016/j.jcp.2022.111513_br0400 article-title: Construction of a pathway map on a complicated energy landscape publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.124.090601 – volume: 51 start-page: 2479 year: 2019 ident: 10.1016/j.jcp.2022.111513_br0040 article-title: Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications publication-title: SIAM J. Math. Anal. doi: 10.1137/18M1224970 – volume: 86 start-page: 29 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0510 article-title: Optimal Petrov-Galerkin spectral approximation method for the fractional diffusion, advection, reaction equation on a bounded interval publication-title: J. Sci. Comput. doi: 10.1007/s10915-020-01366-y – volume: 299 start-page: 196 year: 2015 ident: 10.1016/j.jcp.2022.111513_br0320 article-title: Spectral direction splitting methods for two-dimensional space fractional diffusion equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2015.07.011 – start-page: 1 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0230 article-title: Fast computation of the multidimensional fractional Laplacian publication-title: Appl. Anal. – volume: 98 year: 2007 ident: 10.1016/j.jcp.2022.111513_br0460 article-title: Morphology of critical nuclei in solid-state phase transformations publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.98.265703 – volume: 37 start-page: A2026 year: 2015 ident: 10.1016/j.jcp.2022.111513_br0220 article-title: Deflation techniques for finding distinct solutions of nonlinear partial differential equations publication-title: SIAM J. Sci. Comput. doi: 10.1137/140984798 – volume: 252 start-page: 495 year: 2013 ident: 10.1016/j.jcp.2022.111513_br0440 article-title: Fractional Sturm-Liouville eigen-problems: theory and numerical approximation publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2013.06.031 – volume: 58 start-page: 330 year: 2020 ident: 10.1016/j.jcp.2022.111513_br0500 article-title: An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes publication-title: SIAM J. Numer. Anal. doi: 10.1137/19M1245621 – volume: 116 start-page: 3777 year: 2002 ident: 10.1016/j.jcp.2022.111513_br0130 article-title: Saddle points and dynamics of Lennard-Jones clusters, solids, and supercooled liquids publication-title: J. Chem. Phys. doi: 10.1063/1.1436470 – volume: 29 start-page: 57 year: 2002 ident: 10.1016/j.jcp.2022.111513_br0200 article-title: Variable order and distributed order fractional operators publication-title: Nonlinear Dyn. doi: 10.1023/A:1016586905654 – ident: 10.1016/j.jcp.2022.111513_br0210 – volume: 388 start-page: 4586 year: 2009 ident: 10.1016/j.jcp.2022.111513_br0330 article-title: Variable-order fractional differential operators in anomalous diffusion modeling publication-title: Phys. A, Stat. Mech. Appl. doi: 10.1016/j.physa.2009.07.024 – volume: 19 start-page: 19 year: 2018 ident: 10.1016/j.jcp.2022.111513_br0070 article-title: Numerical methods for fractional diffusion publication-title: Comput. Vis. Sci. doi: 10.1007/s00791-018-0289-y – volume: 36 start-page: 708 year: 2014 ident: 10.1016/j.jcp.2022.111513_br0280 article-title: Two-phase fluid simulation using a diffuse interface model with Peng–Robinson equation of state publication-title: SIAM J. Sci. Comput. doi: 10.1137/130933745 – volume: 42 start-page: A585 year: 2020 ident: 10.1016/j.jcp.2022.111513_br0350 article-title: Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains publication-title: SIAM J. Sci. Comput. doi: 10.1137/19M1244299 – volume: 101 start-page: 1848 year: 2022 ident: 10.1016/j.jcp.2022.111513_br0490 article-title: Variable-order space-fractional diffusion equations and a variable-order modification of constant-order fractional problems publication-title: Appl. Anal. doi: 10.1080/00036811.2020.1789596 – volume: 430 year: 2022 ident: 10.1016/j.jcp.2022.111513_br0430 article-title: Solution landscape of the Onsager model identifies non-axisymmetric critical points publication-title: Phys. D, Nonlin. Phenom. doi: 10.1016/j.physd.2021.133081 – volume: 84 year: 2011 ident: 10.1016/j.jcp.2022.111513_br0250 article-title: Finding all the stationary points of a potential-energy landscape via numerical polynomial-homotopy-continuation method publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.84.025702 – volume: 477 start-page: 20210458 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0190 article-title: Solution landscapes of the simplified Ericksen-Leslie model and its comparison with the reduced Landau-de Gennes model publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2021.0458 – volume: 114 start-page: 93 year: 2019 ident: 10.1016/j.jcp.2022.111513_br0090 article-title: Well-posedness, regularity and asymptotic analyses for a fractional phase field system publication-title: Asymptot. Anal. – volume: 278 start-page: 294 year: 2021 ident: 10.1016/j.jcp.2022.111513_br0170 article-title: Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2020.12.034 |
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