A scalable DG solver for the electroneutral Nernst-Planck equations. (English) Zbl 07649280
Summary: The robust, scalable simulation of flowing electrochemical systems is increasingly important due to the synergy between intermittent renewable energy and electrochemical technologies such as energy storage and chemical manufacturing. The high Péclet regime of many such applications prevents the use of off-the-shelf discretization methods. In this work, we present a high-order Discontinuous Galerkin scheme for the electroneutral Nernst-Planck equations. The chosen charge conservation formulation allows for the specific treatment of the different physics: upwinding for advection and migration, and interior penalty for diffusion of ionic species as well the electric potential. Similarly, the formulation enables different treatments in the preconditioner: AMG for the potential blocks and ILU-based methods for the advection-dominated concentration blocks. We evaluate the convergence rate of the discretization scheme through numerical tests. Strong scaling results for two preconditioning approaches are shown for a large 3D flow-plate reactor example.
MSC:
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
| 65Fxx | Numerical linear algebra |
| 65Yxx | Computer aspects of numerical algorithms |
Keywords:
electrochemical flow; Nernst-Planck; electroneutrality; discontinuous Galerkin; preconditioningReferences:
| [1] | Chu, S.; Majumdar, A., Opportunities and challenges for a sustainable energy future, Nature, 488, 7411, 294-303 (2012) |
| [2] | Chu, S.; Cui, Y.; Liu, N., The path towards sustainable energy, Nat. Mater., 16, 1, 16-22 (2016) |
| [3] | Gür, T. M., Review of electrical energy storage technologies, materials and systems: challenges and prospects for large-scale grid storage, Energy Environ. Sci., 11, 10, 2696-2767 (2018) |
| [4] | Ager, J. W.; Lapkin, A. A., Chemical storage of renewable energy, Science, 360, 6390, 707-708 (2018) |
| [5] | Newman, J.; Thomas-Alyea, K. E., Electrochemical Systems (2012), John Wiley & Sons |
| [6] | Dickinson, E. J.; Limon-Petersen, J. G.; Compton, R. G., The electroneutrality approximation in electrochemistry, J. Solid State Electrochem., 15, 7-8, 1335-1345 (2011) |
| [7] | Bortels, L.; Deconinck, J.; Van Den Bossche, B., The multi-dimensional upwinding method as a new simulation tool for the analysis of multi-ion electrolytes controlled by diffusion, convection and migration. Part 1. Steady state analysis of a parallel plane flow channel, J. Electroanal. Chem., 404, 1, 15-26 (1996) |
| [8] | Bauer, G., A coupled finite element approach for electrochemical systems (2012), Technische Universität München, Ph.D. thesis |
| [9] | Liu, H.; Wang, Z., A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson-Nernst-Planck systems, J. Comput. Phys., 328, 413-437 (2017) · Zbl 1406.82021 |
| [10] | Sun, Z.; Carrillo, J. A.; Shu, C.-W., A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials, J. Comput. Phys., 352, 76-104 (2018) · Zbl 1380.65287 |
| [11] | Hartmann, R.; Leicht, T., Higher order and adaptive DG methods for compressible flows, (37th Advanced CFD Lecture Series: Recent Developments in Higher Order Methods and Industrial Application in Aeronautics 2014 (3) (2014)), 1-156 |
| [12] | Wathen, A. J., Preconditioning, Acta Numer., 24, 329-376 (2015) · Zbl 1316.65039 |
| [13] | Rivière, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation (2008), SIAM · Zbl 1153.65112 |
| [14] | Brezzi, F.; Marini, L. D.; Süli, E., Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14, 12, 1893-1903 (2004) · Zbl 1070.65117 |
| [15] | Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 3, 173-261 (2001) · Zbl 1065.76135 |
| [16] | Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779 (2002) · Zbl 1008.65080 |
| [17] | McRae, A. T.T.; Bercea, G.-T.; Mitchell, L.; Ham, D. A.; Cotter, C. J., Automated generation and symbolic manipulation of tensor product finite elements, SIAM J. Sci. Comput., 38, 5, S25-S47 (2016) · Zbl 1352.65615 |
| [18] | Dennis, J. E.; Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1996), SIAM · Zbl 0847.65038 |
| [19] | Ruge, J. W.; Stüben, K., Algebraic multigrid, (Multigrid Methods. Multigrid Methods, Frontiers in Applied Mathematics, vol. 3 (1987), SIAM: SIAM Philadelphia), 73-130, Ch. 4 |
| [20] | Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018 |
| [21] | Widlund, O.; Dryja, M., An additive variant of the Schwarz alternating method for the case of many subregions, Tech. rep. (1987) |
| [22] | Meijerink, J. A.; van der Vorst, H. A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput., 31, 137, 148-162 (1977) · Zbl 0349.65020 |
| [23] | Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comput., 31, 138, 333-390 (1977) · Zbl 0373.65054 |
| [24] | May, D. A.; Sanan, P.; Rupp, K.; Knepley, M. G.; Smith, B. F., Extreme-scale multigrid components within PETSc, (Proceedings of the Platform for Advanced Scientific Computing Conference, PASC ’16 (2016), Association for Computing Machinery: Association for Computing Machinery New York, NY, USA) |
| [25] | Fidkowski, K. J.; Oliver, T. A.; Lu, J.; Darmofal, D. L., p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, J. Comput. Phys., 207, 1, 92-113 (2005) · Zbl 1177.76194 |
| [26] | Helenbrook, B.; Mavriplis, D.; Atkins, H., Analysis of “p”-multigrid for continuous and discontinuous finite element discretizations, (16th AIAA Computational Fluid Dynamics Conference (2003)), 3989 |
| [27] | Roy, T.; Andrej, J.; Beck, V.; Ehlinger, V.; Govindarajan, N.; Lin, T., LLNL/echemfem: EchemFEM (Dec. 2022) |
| [28] | Rathgeber, F.; Ham, D. A.; Mitchell, L.; Lange, M.; Luporini, F.; McRae, A. T.T.; Bercea, G.-T.; Markall, G. R.; Kelly, P. H.J., Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw., 43, 3, 24:1-24:27 (2016) · Zbl 1396.65144 |
| [29] | Homolya, M.; Ham, D. A., A parallel edge orientation algorithm for quadrilateral meshes, SIAM J. Sci. Comput., 38, 5, S48-S61 (2016) · Zbl 1404.65266 |
| [30] | Bercea, G.; McRae, A. T.T.; Ham, D. A.; Mitchell, L.; Rathgeber, F.; Nardi, L.; Luporini, F.; Kelly, P. H.J., A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake, Geosci. Model Dev., 9, 10, 3803-3815 (2016) |
| [31] | Mitchell, L.; Müller, E. H., High level implementation of geometric multigrid solvers for finite element problems: applications in atmospheric modelling, J. Comput. Phys., 327, 1-18 (2016) · Zbl 1422.65440 |
| [32] | Balay, S.; Gropp, W. D.; McInnes, L. C.; Smith, B. F., Efficient management of parallelism in object oriented numerical software libraries, (Arge, E.; Bruaset, A. M.; Langtangen, H. P., Modern Software Tools in Scientific Computing (1997), Birkhäuser Press), 163-202 · Zbl 0882.65154 |
| [33] | Henson, V. E.; Yang, U. M., BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41, 1, 155-177 (2002) · Zbl 0995.65128 |
| [34] | hypre: High performance preconditioners · Zbl 1056.65046 |
| [35] | Kirby, R. C.; Mitchell, L., Solver composition across the PDE/linear algebra barrier, SIAM J. Sci. Comput., 40, 1, C76-C98 (2018) · Zbl 1383.65021 |
| [36] | Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 2, 461-469 (1993) · Zbl 0780.65022 |
| [37] | Ayuso, B.; Marini, L. D., Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 47, 2, 1391-1420 (2009) · Zbl 1205.65308 |
| [38] | Cockburn, B.; Dong, B.; Guzmán, J., Optimal convergence of the original DG method for the transport-reaction equation on special meshes, SIAM J. Numer. Anal., 46, 3, 1250-1265 (2008) · Zbl 1168.65058 |
| [39] | Richter, G. R., An optimal-order error estimate for the discontinuous Galerkin method, Math. Comput., 50, 181, 75-88 (1988) · Zbl 0643.65059 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
