Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations

Saulo Orizaga; Maurice Fabien; Michael Millard; Saulo Orizaga; Maurice Fabien; Michael Millard

doi:10.3934/math.20241334

AIMS Mathematics

2024, Volume 9, Issue 10: 27471-27496. doi: 10.3934/math.20241334

Previous Article Next Article

Research article

Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations

1.
Department of Mathematics, New Mexico Tech, 801 Leroy Place, Socorro, NM 87801, USA
2.
Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 213,480 Lincoln Dr, Madison, WI 53706, USA

Received: 30 July 2024 Revised: 02 September 2024 Accepted: 13 September 2024 Published: 23 September 2024
MSC : 65M99, 65T50

In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.
- phase-field models,
- Cahn-Hilliard equation,
- thin-film equation,
- efficient numerical methods,
- GPU computation
Citation: Saulo Orizaga, Maurice Fabien, Michael Millard. Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations[J]. AIMS Mathematics, 2024, 9(10): 27471-27496. doi: 10.3934/math.20241334

Related Papers:

Abstract

In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.

References

[1]	J. W. Cahn, J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. interfacial free energy, J.f Chem. Phys., 28 (1958), 258–267. https://doi.org/10.1063/1.1744102 doi: 10.1063/1.1744102
[2]	D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, MRS Online Proc. Libr., 529 (1998), 39–46. https://doi.org/10.1557/PROC-529-39 doi: 10.1557/PROC-529-39
[3]	J. M. Church, Z. Guo, P. K. Jimack, A. Madzvamuse, K. Promislow, B. Wetton, et al., High accuracy benchmark problems for allen-cahn and cahn-hilliard dynamics, Commun. Comput. Phys., 26 (2019), 947–972. https://doi.org/10.4208/cicp.OA-2019-0006 doi: 10.4208/cicp.OA-2019-0006
[4]	Y. Yan, W. Chen, C. Wang, S. M. Wise, A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation, Commun. Comput. Phys., 23 (2018), 572–602. https://doi.org/10.4208/cicp.OA-2016-0197 doi: 10.4208/cicp.OA-2016-0197
[5]	H. Song, Energy SSP-IMEX Runge-Kutta methods for the Cahn-Hilliard equation, J. Comput. Appl. Math., 292 (2016), 576–590. https://doi.org/10.1016/j.cam.2015.07.030 doi: 10.1016/j.cam.2015.07.030
[6]	K. Glasner, S. Orizaga, Improving the accuracy of convexity splitting methods for gradient flow equations, J. Comput. Phys., 315 (2016), 52–64. https://doi.org/10.1016/j.jcp.2016.03.042 doi: 10.1016/j.jcp.2016.03.042
[7]	A. L. Bertozzi, N. Ju, H.-W. Lu, A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations, Discrete Contin. Dyn. Syst., 29 (2011), 1367–1391. https://doi.org/10.3934/dcds.2011.29.1367 doi: 10.3934/dcds.2011.29.1367
[8]	J. Shen, J. Xu, J. Yang, The scalar auxiliary variable (sav) approach for gradient flows, J. Comput. Phys., 353 (2018), 407–416. https://doi.org/10.1016/j.jcp.2017.10.021 doi: 10.1016/j.jcp.2017.10.021
[9]	J. Shen, J. Xu, J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev., 61 (2019), 474–506. https://doi.org/10.1137/17M1150153 doi: 10.1137/17M1150153
[10]	G. Akrivis, B. Li, D. Li, Energy-decaying extrapolated rk–sav methods for the Allen–Cahn and Cahn–Hilliard equations, SIAM J. Sci. Comput., 41 (2019), A3703–A3727. https://doi.org/10.1137/19M1264412 doi: 10.1137/19M1264412
[11]	S. Orizaga, K. Glasner, Instability and reorientation of block copolymer microstructure by imposed electric fields, Phys. Rev. E, 93 (2016), 052504. https://doi.org/10.1103/PhysRevE.93.052504 doi: 10.1103/PhysRevE.93.052504
[12]	K. R. Elder, M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004), 051605. https://doi.org/10.1103/PhysRevE.70.051605 doi: 10.1103/PhysRevE.70.051605
[13]	H. Gomez, X. Nogueira, An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Eng., 249–252 (2012), 52–61. https://doi.org/10.1016/j.cma.2012.03.002 doi: 10.1016/j.cma.2012.03.002
[14]	P. Vignal, L. Dalcin, D. L. Brown, N. Collier, V. M. Calo, An energy-stable convex splitting for the phase-field crystal equation, Comput. Struct., 158 (2015), 355–368. https://doi.org/10.1016/j.compstruc.2015.05.029 doi: 10.1016/j.compstruc.2015.05.029
[15]	Z. Hu, S. M. Wise, C. Wang, J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323–5339. https://doi.org/10.1016/j.jcp.2009.04.020 doi: 10.1016/j.jcp.2009.04.020
[16]	S. M. Wise, C. Wang, J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269–2288. https://doi.org/10.1137/080738143 doi: 10.1137/080738143
[17]	H. Garcke, K. F. Lam, V. Styles, Cahn–hilliard inpainting with the double obstacle potential, SIAM J. Imaging Sci., 11 (2018), 2064–2089. https://doi.org/10.1137/18M1165633 doi: 10.1137/18M1165633
[18]	S. M. Wise, J. S. Lowengrub, H. B. Frieboes, V. Cristini, Three-dimensional multispecies nonlinear tumor growth—Ⅰ: Model and numerical method, J. Theor. Biol., 253 (2008), 524–543.
[19]	V. Cristini, X. Li, J. Lowengrub, S. G. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723–763. https://doi.org/10.1007/s00285-008-0215-x doi: 10.1007/s00285-008-0215-x
[20]	L. N. Trefethen, Spectral methods in MatLab, Philadelphia: Society for Industrial and Applied Mathematics, 2000.
[21]	J. Shen, X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669–1691. https://doi.org/10.3934/dcds.2010.28.1669 doi: 10.3934/dcds.2010.28.1669
[22]	L. Duchemin, J. Eggers, The explicit–implicit–null method: Removing the numerical instability of PDEs, J. Comput. Phys., 263 (2014), 37–52. https://doi.org/10.1016/j.jcp.2014.01.013 doi: 10.1016/j.jcp.2014.01.013
[23]	S. Orizaga, T. Witelski, Imex methods for thin-film equations and cahn–hilliard equations with variable mobility, Comput. Mater. Sci., 243 (2024), 113145. https://doi.org/10.1016/j.commatsci.2024.113145 doi: 10.1016/j.commatsci.2024.113145
[24]	R. R. Rosales, B. Seibold, D. Shirokoff, D. Zhou, Unconditional stability for multistep imex schemes: Theory, SIAM J. Numer. Anal., 55 (2017), 2336–2360. https://doi.org/10.1137/16M1094324 doi: 10.1137/16M1094324
[25]	J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Aust. Math. Soc., 3 (1963), 185–201. https://doi.org/10.1017/S1446788700027932 doi: 10.1017/S1446788700027932
[26]	U. M. Ascher, S. J. Ruuth, B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797–823. https://doi.org/10.1137/0732037 doi: 10.1137/0732037
[27]	H. D. Ceniceros, C. J. García-Cervera, A new approach for the numerical solution of diffusion equations with variable and degenerate mobility, J. Comput. Phys., 246 (2013), 1–10. https://doi.org/10.1016/j.jcp.2013.03.036 doi: 10.1016/j.jcp.2013.03.036
[28]	R. J. LeVeque, Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems, Philadelphia: SIAM, 2007.
[29]	P. J. Roache, The method of manufactured solutions for code verification, In: Computer Simulation Validation. Simulation Foundations, Methods and Applications, Cham: Springer, 2019. https://doi.org/10.1007/978-3-319-70766-2_12
[30]	M. S. Fabien, M. G. Knepley, B. M. Rivière, A hybridizable discontinuous galerkin method for two-phase flow in heterogeneous porous media, Int. J. Numer. Methods Eng., 116 (2018), 161–177. https://doi.org/10.1002/nme.5919 doi: 10.1002/nme.5919
[31]	M. S. Fabien, M. G. Knepley, B. M. Riviere, A high order hybridizable discontinuous galerkin method for incompressible miscible displacement in heterogeneous media, Results Appl. Math., 8 (2020), 100089. https://doi.org/10.1016/j.rinam.2019.100089 doi: 10.1016/j.rinam.2019.100089
[32]	X. Liu, J. Shen, X. Zhang, A simple gpu implementation of spectral-element methods for solving 3d poisson type equations on rectangular domains and its applications, 2024. https://doi.org/10.48550/arXiv.2310.00226
[33]	M. A. Y.-H. Lam, L. J. Cummings, L. Kondic, Computing dynamics of thin films via large scale gpu-based simulations, J. Comput. Phys.: X, 2 (2019), 100001. https://doi.org/10.1016/j.jcpx.2018.100001 doi: 10.1016/j.jcpx.2018.100001
[34]	S. Dai, Q. Du, Computational studies of coarsening rates for the Cahn-Hilliard equation with phase-dependent diffusion mobility, J. Comput. Phys., 310 (2016), 85–108. https://doi.org/10.1016/j.jcp.2016.01.018 doi: 10.1016/j.jcp.2016.01.018
[35]	M. B. Gratton, T. P. Witelski, Coarsening of unstable thin films subject to gravity, Phys. Rev. E, 77 (2008), 016301.
[36]	K. B. Glasner, T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), 016302.
[37]	L. Q. Chen, Phase-field models for microstructure evolution, Ann. Rev. Mater. Res., 32 (2002), 113–140. https://doi.org/10.1146/annurev.matsci.32.112001.132041 doi: 10.1146/annurev.matsci.32.112001.132041
[38]	S. Dai, Q. Du, Coarsening mechanism for systems governed by the Cahn–Hilliard equation with degenerate diffusion mobility, Multiscale Model. Simul., 12 (2014), 1870–1889. https://doi.org/10.1137/140952387 doi: 10.1137/140952387
[39]	C. Zhang, J. Ouyang, C. Wang, S. M. Wise, Numerical comparison of modified-energy stable SAV-type schemes and classical BDF methods on benchmark problems for the functionalized cahn-hilliard equation, J. Comput. Phys., 423 (2020), 109772.
[40]	N. Gavish, J. Jones, Z. Xu, A. Christlieb, K. Promislow, Variational models of network formation and ion transport: Applications to perfluorosulfonate ionomer membranes, Polymers, 4 (2012), 630–655. https://doi.org/10.3390/polym4010630 doi: 10.3390/polym4010630
[41]	A. Oron, S. H. Davis, S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931–980.
[42]	T. P. Witelski, A. J. Bernoff, Stability of self-similar solutions for van der Waals driven thin film rupture, Phys. Fluids, 11 (1999), 2443–2445. https://doi.org/10.1063/1.870138 doi: 10.1063/1.870138
[43]	S. Orizaga, O. Ifeacho, S. Owusu, On an efficient numerical procedure for the Functionalized Cahn-Hilliard equation, AIMS Mathematics, 9 (2024), 20773–20792. https://doi.org/10.3934/math.20241010 doi: 10.3934/math.20241010
[44]	M. W. Noble, M. R. Tonks, S. P. Fitzgerald, Turing instability in the solid state: Void lattices in irradiated metals, Phys. Rev. Lett., 124 (2020), 167401.
[45]	K. Glasner, Segregation and domain formation in non-local multi-species aggregation equations, Phys. D: Nonlinear Phenom., 456 (2023), 133936. https://doi.org/10.1016/j.physd.2023.133936 doi: 10.1016/j.physd.2023.133936

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)