Nonlinear Galerkin methods for a system of PDEs with Turing instabilities. (English) Zbl 1448.65247
Summary: We address and discuss the application of nonlinear Galerkin methods for the model reduction and numerical solution of partial differential equations (PDE) with Turing instabilities in comparison with standard (linear) Galerkin methods. The model considered is a system of PDEs modelling the pattern formation in vegetation dynamics. In particular, by constructing the approximate inertial manifold on the basis of the spectral decomposition of the solution, we implement the so-called Euler-Galerkin method and we compare its efficiency and accuracy versus the linear Galerkin methods. We compare the efficiency of the methods by (a) the accuracy of the computed bifurcation points, and, (b) by the computation of the Hausdorff distance between the limit sets obtained by the Galerkin methods and the ones obtained with a reference finite difference scheme. The efficiency with respect to the required CPU time is also accessed. For our illustrations we used three different ODE time integrators, from the
Matlab ODE suite. Our results indicate that the performance of the Euler-Galerkin method is superior compared to the linear Galerkin method when either explicit or linearly implicit time integration scheme are adopted. For the particular problem considered, we found that the dimension of approximate inertial manifold is strongly affected by the lenght of the spatial domain. Indeeed, we show that the number of modes required to accurately describe the long time Turing pattern forming solutions increases as the domain increases.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 37L65 | Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 35B32 | Bifurcations in context of PDEs |
| 92C80 | Plant biology |
| 92C15 | Developmental biology, pattern formation |
References:
| [1] | Adrover, A., Continillo, G., Crescitelli, S., Giona, M., Russo, L.: Wavelet-like collocation method for finite-dimensional reduction of distributed systems. Comput. Chem. Eng. 24(12), 2687-2703 (2000) · doi:10.1016/S0098-1354(00)00621-9 |
| [2] | Adrover, A., Continillo, G., Crescitelli, S., Gionaa, M., Russo, L.: Construction of approximate inertial manifold by decimation of collocation equations of distributed parameter systems. Comput. Chem. Eng. 26(1), 113-123 (2002) · doi:10.1016/S0098-1354(01)00760-8 |
| [3] | Arrieta, J.M., Santamara, E.: Distance of attractors of reaction-diffusion equations in thin domains. J. Differ. Equ. 263(9), 5459-5506 (2017) · Zbl 1401.35208 · doi:10.1016/j.jde.2017.06.023 |
| [4] | Bizon, K., Continillo, G., Russo, L., Smua, J.: On POD reduced models of tubular reactor with periodic regimes. Comput. Chem. Eng. 32(6), 1305-1315 (2008) · doi:10.1016/j.compchemeng.2007.06.004 |
| [5] | Cartenì, F., Marasco, A., Bonanomi, G., Mazzoleni, S., Rietkerk, M., Giannino, F.: Negative plant soil feedback and ring formation in clonal plants. J. Theor. Biol. 313, 153-161 (2012) · Zbl 1337.92132 · doi:10.1016/j.jtbi.2012.08.008 |
| [6] | Chen, M., Temam, R.: Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns. Numer. Math. 64, 271-294 (1993) · Zbl 0798.65093 · doi:10.1007/BF01388690 |
| [7] | Constantin, P., Foias, C., Nicolaenko, B., Temam, R.: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Springer, Berlin (1989) · Zbl 0683.58002 · doi:10.1007/978-1-4612-3506-4 |
| [8] | Crawford, J.D., Knobloch, E.: On degenerate Hopf bifurcation with broken O(2) symmetry. Nonlinearity 1, 617-652 (1988) · Zbl 0671.34040 · doi:10.1088/0951-7715/1/4/007 |
| [9] | Dettori, L.: Spectral approximations of attractors of a class of semilinear parabolic equations. Galcolo 27, 139-168 (1990) · Zbl 0788.35066 |
| [10] | Devulder, C., Marion, M.: Class of numerical algorithms for large time integration: the nonlinear Galerkin methods. SIAM J. Num. Anal. 29(2), 462-483 (1992) · Zbl 0754.65080 · doi:10.1137/0729028 |
| [11] | Dhooge, A., Govaerts, W., Kuznetsof, Y.A.: MatCont: a matlab package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141-164 (2003) · Zbl 1070.65574 · doi:10.1145/779359.779362 |
| [12] | Dubois, T., Jauberteau, F., Marion, M., Temam, R.: Subgrid modelling and the interaction of small and large wavelengths in turbulent flows. Comput. Phys. Commun. 65(1-3), 100-106 (1991) · Zbl 0900.76082 · doi:10.1016/0010-4655(91)90160-M |
| [13] | Foias, C., Sell, G.R., Temam, R.: Inertial manifolds for nonlinear evolutionary equations. J. Differ. Equ. 73, 309-353 (1988) · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6 |
| [14] | Foias, C., Jolly, M.S., Kevrekidis, I.G., Sell, G.R., Titi, E.S.: On the computation of inertial manifolds. Phys. Lett. A 131(7), 433-437 (1988) · doi:10.1016/0375-9601(88)90295-2 |
| [15] | Garcia-Archilla, B.: Some practical experience with the time integration of dissipative equations. J. Comput. Phys. 122(1), 25-29 (1995) · Zbl 0854.65078 · doi:10.1006/jcph.1995.1193 |
| [16] | Garcia-Archilla, B., Frutos, J.: Time integration of the non-linear Galerkin method. IMA J. Numer. Anal. 15(2), 221-244 (1995) · Zbl 0823.65091 · doi:10.1093/imanum/15.2.221 |
| [17] | Gilad, E., von Hardenberg, J., Provenzale, A., Shachak, M., Meron, E.: Ecosystem engineers: from pattern formation to habitat creation. Phys. Rev. Lett. 93, 1-4 (2004) · doi:10.1103/PhysRevLett.93.098105 |
| [18] | Goubet, O.: Construction of approximate inertial manifolds using wavelets. SIAM J. Math. Anal. 23, 1455-1481 (1992) · Zbl 0770.35003 · doi:10.1137/0523083 |
| [19] | Graham, M.D., Kevrekidis, I.G.: Alternative approaches to the Karhunen-Loeve decomposition for model reduction and data analysis. Comput. Chem. Eng. 20, 495-506 (1996) · doi:10.1016/0098-1354(95)00040-2 |
| [20] | Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A + 2B 3B. B C. Chem. Eng. Sci. 39, 1087-1097 (1984) · doi:10.1016/0009-2509(84)87017-7 |
| [21] | Grosso, M., Russo, L., Maffetone, P.L., Crescitelli, S.: Nonlinear Galerkin method for numerical approximation of the dynamics of mesophases under flow. https://doi.org/10.1109/COC.2000.874332 (2000) · Zbl 0602.58033 |
| [22] | Haken, H.: Synergetics, an Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology. Springer, New York (1983) · Zbl 0523.93001 · doi:10.1007/978-3-642-88338-5 |
| [23] | von Hardenberg, J., Meron, E., Shachak, M., Zarm, I.Y.: Diversity of vegetation patterns and desertification. Phys. Rev. Lett. 87, 198101-4 (2001) · doi:10.1103/PhysRevLett.87.198101 |
| [24] | Heywood, J., Rannacher, R.: On the question of turbulence modeling by approximate inertial manifolds and the nonlinear Galerkin method. SIAM J. Numer. Anal. 30(6), 1603-1621 (1993) · Zbl 0791.76042 · doi:10.1137/0730083 |
| [25] | HilleRisLambers, R., Rietkerk, M., Bosch, F.V.D., Prins, H.H.T., Kroon, H.D.: Vegetation pattern formation in semi-arid grazing systems. Ecology 82, 50-61 (2001) · doi:10.1890/0012-9658(2001)082[0050:VPFISA]2.0.CO;2 |
| [26] | Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996) · Zbl 0890.76001 · doi:10.1017/CBO9780511622700 |
| [27] | Hyman, J.M., Nicolaenko, B.: The Kuramoto-Sivashinsky equation: a bridge between PDEs and dynamical systems. Phys. D 18, 113-126 (1986) · Zbl 0602.58033 · doi:10.1016/0167-2789(86)90166-1 |
| [28] | Jolly, M.S., Kevrekidis, I.G., Titi, E.S.: Approximate inertial manifolds for the Kuramoto-Sivashinski equation: analysis and computations. Phys. D 44, 38-60 (1990) · Zbl 0704.58030 · doi:10.1016/0167-2789(90)90046-R |
| [29] | Jolly, M.S., Rosa, R., Temam, R.: Accurate computations on inertial manifolds. SIAM J. Sci. Comput. 22(6), 2216-2238 (2001) · Zbl 0988.34048 · doi:10.1137/S1064827599351738 |
| [30] | Jones, D.A., Margolin, L.G., Titi, E.S.: On the effectiveness of the approximate inertial manifold a computational study. Theor. Comput. Fluid Dyn. 7, 243-260 (1995) · Zbl 0838.76066 · doi:10.1007/BF00312444 |
| [31] | Kan, X., Duan, J., Kevrekidis, I.G., Roberts, A.J.: Simulating stochastic inertial manifolds by a backward-forward approach. SIAM J. Appl. Dyn. Syst. 12(1), 487-514 (2013) · Zbl 1286.37055 · doi:10.1137/120881968 |
| [32] | Kirby, M.: Minimal dynamical systems from PDEs using sobolev eigenfunctions. Phys. D 57, 466-475 (1992) · Zbl 0760.35041 · doi:10.1016/0167-2789(92)90014-E |
| [33] | Klausmeier, C.A.: Regular and irregular patterns in semiarid vegetation. Science 284, 1826-8 (1999) · doi:10.1126/science.284.5421.1826 |
| [34] | Lord, G.J.: Attractors and inertial manifolds for finite-difference approximations of the complex Ginzburg-Landau equation. SIAM J. Num. Anal. 34(4), 1483-1512 (1997) · Zbl 0888.65104 · doi:10.1137/S003614299528554X |
| [35] | Lu, F., Lin, K.K., Chorin, A.J.: Data-based stochastic model reduction for the Kuramoto-Sivashinsky equation. Phys. D 340(1), 46-57 (2017) · Zbl 1376.35100 · doi:10.1016/j.physd.2016.09.007 |
| [36] | Lunasin, E., Titi, E.S.: Finite determining parameters feedback control for distributed nonlinear dissipative systems a computational study. Evol. Equ. Control Theory 6(4), 535-557 (2017) · Zbl 1375.35256 · doi:10.3934/eect.2017027 |
| [37] | Mach, J., Bene, M., Strachota, P.: Nonlinear Galerkin finite element method applied to the system of reaction diffusion equations in one space dimension. Comput. Math. Appl. 73(9), 2053-2065 (2017) · Zbl 1373.65071 · doi:10.1016/j.camwa.2017.02.032 |
| [38] | Marasco, A., Iuorio, A., Carten, F., Bonanomi, G., Tartakovsky, D., Mazzoleni, S., Giannino, F.: Vegetation pattern formation due to interactions between water availability and toxicity in plant-soil feedback. Bull. Math. Biol. 76, 2866-2883 (2014) · Zbl 1329.92027 · doi:10.1007/s11538-014-0036-6 |
| [39] | Margolin, L.G., Titi, E.S., Wynne, S.: The postprocessing Galerkin and nonlinear Galerkin methods—A truncation analysis point of view. SIAM J. Num. Anal. 41(2), 695-714 (2003) · Zbl 1130.65314 · doi:10.1137/S0036142901390500 |
| [40] | Marion, M., Temam, M.: Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26(5), 11391157 (1989) · Zbl 0683.65083 · doi:10.1137/0726063 |
| [41] | Meinhardt, H.: Models of Biological Pattern Formation. Academic Press, Cambridge (1982) |
| [42] | Meinhardt, H.: The Algorithmic Beauty of Sea Shells. Springer, Berlin (1995) · Zbl 1011.00506 · doi:10.1007/978-3-662-13135-0 |
| [43] | Mengers, J.D., Powers, J.M.: One-dimensional slow invariant manifolds for fully coupled reaction and micro-scale diffusion. SIAM J. Appl. Dyn. Syst. 12(2), 560-595 (2013) · Zbl 1282.35044 · doi:10.1137/120877118 |
| [44] | Meron, E., Gilad, E., von Hardenberg, J., Shachak, M., Zarmi, Y.: Vegetation patterns along a rainfall gradient. Chaos, Solitons & Fractals 19, 367-376 (2004) · Zbl 1083.92039 · doi:10.1016/S0960-0779(03)00049-3 |
| [45] | Nicolaenko, B., Foias, C., Temam, R.: The connection between infinite dimensional and finite dimensional dynamical systems. In: Proceedings of the AMs-IMS-SIAM Joint Summer Research Conference, Contemporary Mathematics series (1989) · Zbl 0681.00010 |
| [46] | Pearson, J.E.: Complex patterns in a simple system. Science 261, 189-192 (1993) · doi:10.1126/science.261.5118.189 |
| [47] | Rietkerk, M., Boerlijst, M.C., van Langevelde, F., Hillerislambers, R., van de Koppel, J., Kumar, L., Prins, H.H.T., de Roos, A.M.: Self-organization of vegetation in arid ecosystems. Am. Nat. 160, 524530 (2002) |
| [48] | Rietkerk, M., Dekker, S.C., de Ruiter, P.C., van de Koppel, J.: Self-organized patchiness and catastrophic shifts in ecosystems. Science 305, 1926-1929 (2004) · doi:10.1126/science.1101867 |
| [49] | Robinson, J.C.: Finite dimensional behavior in dissipative partial differential equations. Chaos 5, 330-345 (1995) · Zbl 1055.35501 · doi:10.1063/1.166081 |
| [50] | Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001) · Zbl 1026.37500 · doi:10.1007/978-94-010-0732-0 |
| [51] | Russo, L., Adrover, A., Continillo, G., Crescitelli, S., Giona, M.: Dynamic behavior of a reaction/diffusion system: wavelet-like collocations and approximate inertial manifolds. Proc. Int. Conf. Control Oscil. Chaos 2, 356-359 (2000) |
| [52] | Scheffer, M.: Critical Transitions in Nature and Society. Princeton University Press, Princeton (2009) |
| [53] | Scheffer, M., Carpenter, S., Foley, J., Folke, C., Walker, B.: Catastrophic shifts in ecosystems. Nature 413, 591-596 (2001) · doi:10.1038/35098000 |
| [54] | Schmidtmann, O., Fuede, F., Seehafer, N.: Non linear Galegrkin methods for 3D magneto-hydrodynamic equations. Int. J. Bifurc. Chaos 7, 1497-1507 (1997) · Zbl 0906.76068 · doi:10.1142/S0218127497001187 |
| [55] | Sembera, J., Bene, M.: Nonlinear Galerkin method for reaction diffusion systems admitting invariant regions. J. Comput. Appl. Math. 136, 163-176 (2001) · Zbl 0990.65112 · doi:10.1016/S0377-0427(00)00582-3 |
| [56] | Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18, 1-22 (1997) · Zbl 0868.65040 · doi:10.1137/S1064827594276424 |
| [57] | Shen, J., Temam, R.: Nonlinear Galerkin method using Chebyshev and Legendre polynomials I. The one-dimensional case. SIAM J. Numer. Anal. 32, 215-234 (1989) · Zbl 0819.35008 · doi:10.1137/0732007 |
| [58] | Sherratt, J.A., Lord, G.J.: Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments. Theor. Popul. Biol. 71, 1-11 (2007) · Zbl 1118.92064 · doi:10.1016/j.tpb.2006.07.009 |
| [59] | Sirovich, L., Knight, B.W., Rodriguez, J.D.: Optimal low-dimensional dynamical approximations. Quart. Appl. Math. XLVIII, 535-548 (1990) · Zbl 0715.58023 · doi:10.1090/qam/1074969 |
| [60] | Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, Berlin (1997) · Zbl 0871.35001 · doi:10.1007/978-1-4612-0645-3 |
| [61] | Temam, R.: Inertial manifolds and multigrid methods. SIAM J. Math. Anal. 21, 154-178 (1990) · Zbl 0715.35039 · doi:10.1137/0521009 |
| [62] | Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237(641), 37-72 (1952) · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012 |
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.
