The Advection-Diffusion Reaction (ADR) equation appears in many problems in nature. This constitutes a general model that is useful in various scenarios, from porous media to atmospheric processes. Particularly, it is used at the interface between two fluids where different types of instabilities due to surface mobility may appear. Together with the ADR equation, the Darcy-Brinkman model describes the phenomena known as fingering that appear in different contexts. The study of this type of system gains in complexity when the number of chemical species dissolved in both fluids increases. With more solutes, the increasing complexity of this phenomenon generally requires much computational power. To face the need for more computational resources, we build a solver tool based on an Alternating Direction Implicit (ADI) scheme that can be run in Central Processing Unit (CPU) and Graphic Processing Unit (GPU) architectures on any notebook. The implementation is done using the MATLAB platform to compare both versions. It is shown that using the GPU version strongly saves both resources and calculation times.
References
[1]
Perevoznikov, E. and Lomteva, E. (2019) Modeling of Economic Processes, Instability and Chaos. Journal of Applied Mathematics and Physics, 7, 356-363. https://doi.org/10.4236/jamp.2019.72027
[2]
Sesini, P.A., de Souza, D.A. and Coutinho, A.L. (2010) Finite Element Simulation of Viscous Fingering in Miscible Displacements at High Mobility-Ratios. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 32, 292-299. https://doi.org/10.1590/S1678-58782010000300013
[3]
Narang, H., Wu, F. and Mohammed, A.R. (2019) An Efficient Acceleration of Solving Heat and Mass Transfer Equations with the First Kind Boundary Conditions in Capillary Porous Radially Composite Cylinder Using Programmable Graphics Hardware. Journal of Computer and Communications, 7, 267-281. https://doi.org/10.4236/jcc.2019.77022
[4]
Zuberek, W.M. (2018) Timed Petri Net Models of Shared-Memory Bus-Based Multiprocessors. Journal of Computer and Communications, 6, 1-14. https://doi.org/10.4236/jcc.2018.610001
[5]
Egidi, N., Giacomini, J., Maponi, P., Perticarini, A., Cognigni, L. and Fioretti, L. (2022) An Advection-Diffusion-Reaction Model for Coffee Percolation. Computational and Ap-plied Mathematics, 41, Article No. 229. https://doi.org/10.1007/s40314-022-01929-9
[6]
Carlotto, T., da Silva, R.V. and Grzybowski, J.M.V. (2019) GPGPU-Accelerated Environmental Modelling Based on the 2D Advection-Reaction-Diffusion Equation. Environmental Modelling & Software, 116, 87-99. https://doi.org/10.1016/j.envsoft.2019.02.001
[7]
Le, P.V., Kumar, P., Valocchi, A.J. and Dang, H.V. (2015) GPU-Based High-Performance Computing for Integrated Surface-Sub-Surface Flow Modeling. Environmental Modelling & Software, 73, 1-13. https://doi.org/10.1016/j.envsoft.2015.07.015
[8]
Su, D., Mayer, K.U. and MacQuarrie, K.T.B. (2017) Parallelization of MIN3P-THCm: A High Performance Computational Framework for Subsurface Flow and Reactive Transport Simulation. Environmental Modelling & Software, 95, 271-289. https://doi.org/10.1016/j.envsoft.2017.06.008
[9]
Li, T., Wang, G., Chen, J. and Wang, H. (2011) Dynamic Parallelization of Hydrological Model Simulations. Environmental Modelling & Software, 26, 1736-1746. https://doi.org/10.1016/j.envsoft.2011.07.015
[10]
Haque, M.N. and Uddin, M.S. (2011) Accelerating Fast Fourier Transformation for Image Processing Using Graphics Processing Unit. International Conference on Signal Processing, Image Processing, and Pattern Recognition, Jeju Island, 8-10 December 2011, 300-309. https://doi.org/10.1007/978-3-642-27183-0_32
[11]
MATLAB, 9.2.0.538062. (R2017a) The MathWorks Inc., Natick, MA.
[12]
Fernandez, D., Binda, L., Zalts, A., El Hasi, C. and D’Onofrio, A. (2018) Lateral Movements in Rayleigh-Taylor Instabilities Due to Frontiers. Numerical Analysis. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28, Article ID: 013108. https://doi.org/10.1063/1.4995396
[13]
Binda, L., Fernandez, D., El Hasi, C., Zalts, A. and D’Onofrio, A. (2018) Lateral Movements in Rayleigh-Taylor Instabilities Due to Frontiers. Experimental Study. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28, Article ID: 013107. https://doi.org/10.1063/1.4995395
[14]
Kuster, S., Riolfo, L.A., Zalts, A., El Hasi, C., Almarcha, C., Trevelyan, P.M.J. and D’Onofrio, A. (2011) Differential Diffusion Effects on Buoyancy-Driven Instabilities of Acid-Base Fronts: The Case of a Color Indicator. Physical Chemistry Chemical Physics, 13, 17295-17303. https://doi.org/10.1039/c1cp21185d
[15]
Whitaker, N. (1990) Numerical Solution of the Hele-Shaw Equations. Journal of Computational Physics, 90, 176-199. https://doi.org/10.1016/0021-9991(90)90202-C http://www.sciencedirect.com/science/article/pii/002199919090202C
[16]
Whitaker, N. (1994) Some Numerical Methods for the Hele-Shaw Equations. Journal of Computational Physics, 111, 81-88. https://doi.org/10.1006/jcph.1994.1046
[17]
Turner, J. (1974) Double-Diffusive Phenomena. Annual Review of Fluid Mechanics, 6, 37-54. https://doi.org/10.1146/annurev.fl.06.010174.000345
[18]
Homsy, G.M. (1987) Viscous Fingering in Porous Media. Annual Review of Fluid Mechanics, 19, 271-311. https://doi.org/10.1146/annurev.fl.19.010187.001415
[19]
Sharp, D.H. (1984) An Overview of Rayleigh-Taylor Instability. Physica D: Nonlinear Phenomena, 12, 3-18. https://doi.org/10.1016/0167-2789(84)90510-4
[20]
Lemaigre, L., Budroni, M.A., Riolfo, L.A., Grosfils, P. and De Wit, A. (2013) Asymmetric Rayleigh-Taylor and Double-Diffusive Fingers in Reactive Systems. Physics of Fluids, 25, Article ID: 014103. https://doi.org/10.1063/1.4774321
[21]
You, Y. (2002) A Global Ocean Climatological Atlas of the Turner Angle: Implications for Double-Diffusion and Water-Mass Structure. Deep Sea Research Part I: Oceanographic Research Papers, 49, 2075-2093. https://doi.org/10.1016/S0967-0637(02)00099-7
[22]
Trevelyan, P.M.J., Almarcha, C. and De Wit, A. (2015) Buoyancy-Driven Instabilities Around Miscible A + B → C Reaction Fronts: A General Classification. Physical Review E, 91, Article ID: 023001. https://doi.org/10.1103/PhysRevE.91.023001
[23]
Zalts, A., El Hasi, C., Rubio, D., Urena, A. and D’Onofrio, A. (2008) Pattern Formation Driven by an Acid-Base Neutralization Reaction in Aqueous Media in a Gravitational Field. Physical Review E, 77, Article ID: 015304. https://doi.org/10.1103/PhysRevE.77.015304
[24]
Tan, C.T. and Homsy, G.M. (1986) Stability of Miscible Displacements in Porous Media: Rectilinear Flow. The Physics of Fluids, 29, Article No. 3548. https://doi.org/10.1063/1.865832
[25]
Mangiavacchi, N., Coutinho, A.L.G.A. and Ebecken, N.F.F. (1997) Parallel Pseudo-Spectral Simulations of Nonlinear Viscous Fingering in Mis-Cible Displacements. WIT Transactions on the Built Environment, 32, 498-506.
[26]
Tan, C.T. and Homsy, G.M. (1988) Simulation of Nonlinear Viscous Fingering in Miscible Displacement. The Physics of Fluids, 31, 1330-1338. https://doi.org/10.1063/1.866726
[27]
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1993) Numerical Recipes in Fortran 77: The Art of Scientific Computing. 2nd Edition, Cambridge University Press, Cambridge, MA.
[28]
Woods, L.C. (1954) A Note on the Numerical Solution of Fourth Order Differential Equations. Aeronautical Quarterly, 5, 176-184. https://doi.org/10.1017/S0001925900001177
[29]
Kim, M.C. and Cardoso, S.S. (2019) Diffusivity Ratio Effect on the Onset of the Buoyancy-Driven Instability of an A + B → C Chemical Reaction System in a Hele-Shaw Cell: Numerical Simulations and Comparison with Experiments. Physics of Fluids, 31, Article ID: 084101. https://doi.org/10.1063/1.5094913
