Coherent structures are spatially varying regions which disperse minimally over time and organise motion in non-autonomous systems. This work develops and implements algorithms providing multilayered descriptions of time-dependent systems which are not only useful for locating coherent structures, but also for detecting time windows within which these structures undergo fundamental structural changes, such as merging and splitting events. These algorithms rely on singular value decompositions associated to Ulam type discretisations of transfer operators induced by dynamical systems, and build on recent developments in multiplicative ergodic theory. Furthermore, they allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. The approach is tested on models of periodically and quasi-periodically driven systems, as well as on a geophysical dataset corresponding to the splitting of the Southern Polar Vortex.
{"title":"A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems","authors":"Chantelle Blachut, C. Gonz'alez-Tokman","doi":"10.3934/jcd.2020015","DOIUrl":"https://doi.org/10.3934/jcd.2020015","url":null,"abstract":"Coherent structures are spatially varying regions which disperse minimally over time and organise motion in non-autonomous systems. This work develops and implements algorithms providing multilayered descriptions of time-dependent systems which are not only useful for locating coherent structures, but also for detecting time windows within which these structures undergo fundamental structural changes, such as merging and splitting events. These algorithms rely on singular value decompositions associated to Ulam type discretisations of transfer operators induced by dynamical systems, and build on recent developments in multiplicative ergodic theory. Furthermore, they allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. The approach is tested on models of periodically and quasi-periodically driven systems, as well as on a geophysical dataset corresponding to the splitting of the Southern Polar Vortex.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82155174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that any system of ODEs can be modified whilst preserving its homogeneous Darboux polynomials. We employ the result to generalise a hierarchy of integrable Lotka-Volterra systems.
{"title":"Homogeneous darboux polynomials and generalising integrable ODE systems","authors":"Peter H. van der Kamp, D. McLaren, G. Quispel","doi":"10.3934/jcd.2021001","DOIUrl":"https://doi.org/10.3934/jcd.2021001","url":null,"abstract":"We show that any system of ODEs can be modified whilst preserving its homogeneous Darboux polynomials. We employ the result to generalise a hierarchy of integrable Lotka-Volterra systems.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77381583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Pozharskiy, Noah J. Wichrowski, A. Duncan, G. Pavliotis, I. Kevrekidis
Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the "curse of dimensionality," becoming ineffective as the dimension of the parameter space grows. One feature of a subclass of such problems that are effectively low-dimensional is that only a few parameters (or combinations thereof) are important for the optimization and must be explored in detail. Knowing these parameters/ combinations in advance would greatly simplify the problem and its solution. We propose the data-driven construction of an effective (coarse-grained, "trend") optimizer, based on data obtained from ensembles of brief simulation bursts with an "inner" optimization algorithm, that has the potential to accelerate the exploration of the parameter space. The trajectories of this "effective optimizer" quickly become attracted onto a slow manifold parameterized by the few relevant parameter combinations. We obtain the parameterization of this low-dimensional, effective optimization manifold on the fly using data mining/manifold learning techniques on the results of simulation (inner optimizer iteration) burst ensembles and exploit it locally to "jump" forward along this manifold. As a result, we can bias the exploration of the parameter space towards the few, important directions and, through this "wrapper algorithm," speed up the convergence of traditional optimization algorithms.
{"title":"Manifold learning for accelerating coarse-grained optimization","authors":"D. Pozharskiy, Noah J. Wichrowski, A. Duncan, G. Pavliotis, I. Kevrekidis","doi":"10.3934/jcd.2020021","DOIUrl":"https://doi.org/10.3934/jcd.2020021","url":null,"abstract":"Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the \"curse of dimensionality,\" becoming ineffective as the dimension of the parameter space grows. One feature of a subclass of such problems that are effectively low-dimensional is that only a few parameters (or combinations thereof) are important for the optimization and must be explored in detail. Knowing these parameters/ combinations in advance would greatly simplify the problem and its solution. We propose the data-driven construction of an effective (coarse-grained, \"trend\") optimizer, based on data obtained from ensembles of brief simulation bursts with an \"inner\" optimization algorithm, that has the potential to accelerate the exploration of the parameter space. The trajectories of this \"effective optimizer\" quickly become attracted onto a slow manifold parameterized by the few relevant parameter combinations. We obtain the parameterization of this low-dimensional, effective optimization manifold on the fly using data mining/manifold learning techniques on the results of simulation (inner optimizer iteration) burst ensembles and exploit it locally to \"jump\" forward along this manifold. As a result, we can bias the exploration of the parameter space towards the few, important directions and, through this \"wrapper algorithm,\" speed up the convergence of traditional optimization algorithms.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90687799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier–Stokes–Cahn–Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and keeping the component concentration inside the physically reasonable range begin{document}$ (0,1) $end{document} . This is achieved, first, by discretizing the non-divergent potential form of terms responsible for the capillary effects and establishing the dissipativity of the discrete full energy. Second, a logarithmic (or the Flory–Huggins potential) form for the non-convex bulk free energy is used. The regularization of equations is accomplished to increase essentially the time step of the explicit discretization in time. We include 3D numerical results for two typical problems that confirm the theoretical predictions.
We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier–Stokes–Cahn–Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and keeping the component concentration inside the physically reasonable range begin{document}$ (0,1) $end{document} . This is achieved, first, by discretizing the non-divergent potential form of terms responsible for the capillary effects and establishing the dissipativity of the discrete full energy. Second, a logarithmic (or the Flory–Huggins potential) form for the non-convex bulk free energy is used. The regularization of equations is accomplished to increase essentially the time step of the explicit discretization in time. We include 3D numerical results for two typical problems that confirm the theoretical predictions.
{"title":"An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations","authors":"V. Balashov, A. Zlotnik","doi":"10.3934/jcd.2020012","DOIUrl":"https://doi.org/10.3934/jcd.2020012","url":null,"abstract":"We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier–Stokes–Cahn–Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and keeping the component concentration inside the physically reasonable range begin{document}$ (0,1) $end{document} . This is achieved, first, by discretizing the non-divergent potential form of terms responsible for the capillary effects and establishing the dissipativity of the discrete full energy. Second, a logarithmic (or the Flory–Huggins potential) form for the non-convex bulk free energy is used. The regularization of equations is accomplished to increase essentially the time step of the explicit discretization in time. We include 3D numerical results for two typical problems that confirm the theoretical predictions.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86396114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mojtaba F. Fathi, Ahmadreza Baghaie, A. Bakhshinejad, Raphael H. Sacho, Roshan M. D'Souza
In this research, we investigate the application of Dynamic Mode Decomposition combined with Kalman Filtering, Smoothing, and Wavelet Denoising (DMD-KF-W) for denoising time-resolved data. We also compare the performance of this technique with state-of-the-art denoising methods such as Total Variation Diminishing (TV) and Divergence-Free Wavelets (DFW), when applicable. Dynamic Mode Decomposition (DMD) is a data-driven method for finding the spatio-temporal structures in time series data. In this research, we use an autoregressive linear model resulting from applying DMD to the time-resolved data as a predictor in a Kalman Filtering-Smoothing framework for the purpose of denoising. The DMD-KF-W method is parameter-free and runs autonomously. Tests on numerical phantoms show lower error metrics when compared to TV and DFW, when applicable. In addition, DMD-KF-W runs an order of magnitude faster than DFW and TV. In the case of synthetic datasets, where the noise-free datasets were available, our method was shown to perform better than TV and DFW methods (when applicable) in terms of the defined error metric.
{"title":"Time-resolved denoising using model order reduction, dynamic mode decomposition, and kalman filter and smoother","authors":"Mojtaba F. Fathi, Ahmadreza Baghaie, A. Bakhshinejad, Raphael H. Sacho, Roshan M. D'Souza","doi":"10.3934/jcd.2020019","DOIUrl":"https://doi.org/10.3934/jcd.2020019","url":null,"abstract":"In this research, we investigate the application of Dynamic Mode Decomposition combined with Kalman Filtering, Smoothing, and Wavelet Denoising (DMD-KF-W) for denoising time-resolved data. We also compare the performance of this technique with state-of-the-art denoising methods such as Total Variation Diminishing (TV) and Divergence-Free Wavelets (DFW), when applicable. Dynamic Mode Decomposition (DMD) is a data-driven method for finding the spatio-temporal structures in time series data. In this research, we use an autoregressive linear model resulting from applying DMD to the time-resolved data as a predictor in a Kalman Filtering-Smoothing framework for the purpose of denoising. The DMD-KF-W method is parameter-free and runs autonomously. Tests on numerical phantoms show lower error metrics when compared to TV and DFW, when applicable. In addition, DMD-KF-W runs an order of magnitude faster than DFW and TV. In the case of synthetic datasets, where the noise-free datasets were available, our method was shown to perform better than TV and DFW methods (when applicable) in terms of the defined error metric.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88342884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by begin{document}$ N times N $end{document} matrices for large enough begin{document}$ N $end{document} . We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.
This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by begin{document}$ N times N $end{document} matrices for large enough begin{document}$ N $end{document} . We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.
{"title":"A functional analytic approach to validated numerics for eigenvalues of delay equations","authors":"J. Lessard, J. D. M. James","doi":"10.3934/jcd.2020005","DOIUrl":"https://doi.org/10.3934/jcd.2020005","url":null,"abstract":"This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by begin{document}$ N times N $end{document} matrices for large enough begin{document}$ N $end{document} . We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78526883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. E. Pérez-López, D. A. Rueda-Gómez, É. J. Villamizar-Roa
This paper is devoted to the theoretical and numerical analysis of the non-stationary Rayleigh-Benard-Marangoni (RBM) system. We analyze the existence of global weak solutions for the non-stationary RBM system in polyhedral domains of begin{document}$ mathbb{R}^3 $end{document} and the convergence, in the norm of begin{document}$ L^{2}(Omega), $end{document} to the corresponding stationary solution. Additionally, we develop a numerical scheme for approximating the weak solutions of the non-stationary RBM system, based on a mixed approximation: finite element approximation in space and finite differences in time. After proving the unconditional well-posedness of the numerical scheme, we analyze some error estimates and establish a convergence analysis. Finally, we present some numerical simulations to validate the behavior of our scheme.
This paper is devoted to the theoretical and numerical analysis of the non-stationary Rayleigh-Benard-Marangoni (RBM) system. We analyze the existence of global weak solutions for the non-stationary RBM system in polyhedral domains of begin{document}$ mathbb{R}^3 $end{document} and the convergence, in the norm of begin{document}$ L^{2}(Omega), $end{document} to the corresponding stationary solution. Additionally, we develop a numerical scheme for approximating the weak solutions of the non-stationary RBM system, based on a mixed approximation: finite element approximation in space and finite differences in time. After proving the unconditional well-posedness of the numerical scheme, we analyze some error estimates and establish a convergence analysis. Finally, we present some numerical simulations to validate the behavior of our scheme.
{"title":"On the Rayleigh-Bénard-Marangoni problem: Theoretical and numerical analysis","authors":"J. E. Pérez-López, D. A. Rueda-Gómez, É. J. Villamizar-Roa","doi":"10.3934/jcd.2020006","DOIUrl":"https://doi.org/10.3934/jcd.2020006","url":null,"abstract":"This paper is devoted to the theoretical and numerical analysis of the non-stationary Rayleigh-Benard-Marangoni (RBM) system. We analyze the existence of global weak solutions for the non-stationary RBM system in polyhedral domains of begin{document}$ mathbb{R}^3 $end{document} and the convergence, in the norm of begin{document}$ L^{2}(Omega), $end{document} to the corresponding stationary solution. Additionally, we develop a numerical scheme for approximating the weak solutions of the non-stationary RBM system, based on a mixed approximation: finite element approximation in space and finite differences in time. After proving the unconditional well-posedness of the numerical scheme, we analyze some error estimates and establish a convergence analysis. Finally, we present some numerical simulations to validate the behavior of our scheme.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84761360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Finite-Time Lyapunov Exponent (FTLE) is a well-established numerical tool for assessing stretching rates of initial parcels of fluid, which are advected according to a given time-varying velocity field (which is often available only as data). When viewed as a field over initial conditions, the FTLE's spatial structure is often used to infer the nonhomogeneous transport. Given the measurement and resolution errors inevitably present in the unsteady velocity data, the computed FTLE field should in reality be treated only as an approximation. A method which, for the first time, is able for attribute spatially-varying errors to the FTLE field is developed. The formulation is, however, confined to two-dimensional flows. Knowledge of the errors prevent reaching erroneous conclusions based only on the FTLE field. Moreover, it is established that increasing the spatial resolution does not improve the accuracy of the FTLE field in the presence of velocity uncertainties, and indeed has the opposite effect. Stochastic simulations are used to validate and exemplify these results, and demonstrate the computability of the error field.
{"title":"Uncertainty in finite-time Lyapunov exponent computations","authors":"Sanjeeva Balasuriya","doi":"10.3934/jcd.2020013","DOIUrl":"https://doi.org/10.3934/jcd.2020013","url":null,"abstract":"The Finite-Time Lyapunov Exponent (FTLE) is a well-established numerical tool for assessing stretching rates of initial parcels of fluid, which are advected according to a given time-varying velocity field (which is often available only as data). When viewed as a field over initial conditions, the FTLE's spatial structure is often used to infer the nonhomogeneous transport. Given the measurement and resolution errors inevitably present in the unsteady velocity data, the computed FTLE field should in reality be treated only as an approximation. A method which, for the first time, is able for attribute spatially-varying errors to the FTLE field is developed. The formulation is, however, confined to two-dimensional flows. Knowledge of the errors prevent reaching erroneous conclusions based only on the FTLE field. Moreover, it is established that increasing the spatial resolution does not improve the accuracy of the FTLE field in the presence of velocity uncertainties, and indeed has the opposite effect. Stochastic simulations are used to validate and exemplify these results, and demonstrate the computability of the error field.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74536396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe a renormalization method in maps of the plane begin{document}$ (x, y) $end{document} , with constant Jacobian begin{document}$ b $end{document} and a second parameter begin{document}$ a $end{document} acting as a bifurcation parameter. The method enables one to organize high period periodic attractors and thus find hordes of them in quasi-conservative maps (i.e. begin{document}$ |b| = 1-varepsilon $end{document} ), when sharing the same rotation number. Numerical challenges are the high period, and the necessary extreme vicinity of many such different points, which accumulate on a hyperbolic periodic saddle. The periodic points are organized, in the begin{document}$ (x, y, a) $end{document} space, in sequences of diverging period, that we call "branches". We define a renormalization approach, by "hopping" among branches to maximize numerical convergence. Our numerical renormalization has met two kinds of numerical instabilities, well localized in certain ranges of the period for the parameter begin{document}$ a $end{document} (see [ 3 ]) and in other ranges of the period for the dynamical plane begin{document}$ (x, y) $end{document} . For the first time we explain here how specific numerical instabilities depend on geometry displacements in dynamical plane begin{document}$ (x, y) $end{document} . We describe how to take advantage of such displacement in the sequence, and of the high period, by moving forward from one branch to its image under dynamics. This, for high period, allows entering the hyperbolicity neighborhood of a saddle, where the dynamics is conjugate to a hyperbolic linear map. The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [ 7 ] for highly dissipative systems.
We describe a renormalization method in maps of the plane begin{document}$ (x, y) $end{document} , with constant Jacobian begin{document}$ b $end{document} and a second parameter begin{document}$ a $end{document} acting as a bifurcation parameter. The method enables one to organize high period periodic attractors and thus find hordes of them in quasi-conservative maps (i.e. begin{document}$ |b| = 1-varepsilon $end{document} ), when sharing the same rotation number. Numerical challenges are the high period, and the necessary extreme vicinity of many such different points, which accumulate on a hyperbolic periodic saddle. The periodic points are organized, in the begin{document}$ (x, y, a) $end{document} space, in sequences of diverging period, that we call "branches". We define a renormalization approach, by "hopping" among branches to maximize numerical convergence. Our numerical renormalization has met two kinds of numerical instabilities, well localized in certain ranges of the period for the parameter begin{document}$ a $end{document} (see [ 3 ]) and in other ranges of the period for the dynamical plane begin{document}$ (x, y) $end{document} . For the first time we explain here how specific numerical instabilities depend on geometry displacements in dynamical plane begin{document}$ (x, y) $end{document} . We describe how to take advantage of such displacement in the sequence, and of the high period, by moving forward from one branch to its image under dynamics. This, for high period, allows entering the hyperbolicity neighborhood of a saddle, where the dynamics is conjugate to a hyperbolic linear map. The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [ 7 ] for highly dissipative systems.
{"title":"A numerical renormalization method for quasi–conservative periodic attractors","authors":"C. Falcolini, Laura Tedeschini-Lalli","doi":"10.3934/jcd.2020018","DOIUrl":"https://doi.org/10.3934/jcd.2020018","url":null,"abstract":"We describe a renormalization method in maps of the plane begin{document}$ (x, y) $end{document} , with constant Jacobian begin{document}$ b $end{document} and a second parameter begin{document}$ a $end{document} acting as a bifurcation parameter. The method enables one to organize high period periodic attractors and thus find hordes of them in quasi-conservative maps (i.e. begin{document}$ |b| = 1-varepsilon $end{document} ), when sharing the same rotation number. Numerical challenges are the high period, and the necessary extreme vicinity of many such different points, which accumulate on a hyperbolic periodic saddle. The periodic points are organized, in the begin{document}$ (x, y, a) $end{document} space, in sequences of diverging period, that we call \"branches\". We define a renormalization approach, by \"hopping\" among branches to maximize numerical convergence. Our numerical renormalization has met two kinds of numerical instabilities, well localized in certain ranges of the period for the parameter begin{document}$ a $end{document} (see [ 3 ]) and in other ranges of the period for the dynamical plane begin{document}$ (x, y) $end{document} . For the first time we explain here how specific numerical instabilities depend on geometry displacements in dynamical plane begin{document}$ (x, y) $end{document} . We describe how to take advantage of such displacement in the sequence, and of the high period, by moving forward from one branch to its image under dynamics. This, for high period, allows entering the hyperbolicity neighborhood of a saddle, where the dynamics is conjugate to a hyperbolic linear map. The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [ 7 ] for highly dissipative systems.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78172039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explore a new asymptotic-numerical solver for the time-dependent wave equation with an interaction term that is oscillating in time with a very high frequency. The method involves representing the solution as an asymptotic series in inverse powers of the oscillation frequency. Using the new scheme, high accuracy is achieved at a low computational cost. Salient features of the new approach are highlighted by a numerical example.
{"title":"Solving the wave equation with multifrequency oscillations","authors":"M. Condon, A. Iserles, K. Kropielnicka, P. Singh","doi":"10.3934/jcd.2019012","DOIUrl":"https://doi.org/10.3934/jcd.2019012","url":null,"abstract":"We explore a new asymptotic-numerical solver for the time-dependent wave equation with an interaction term that is oscillating in time with a very high frequency. The method involves representing the solution as an asymptotic series in inverse powers of the oscillation frequency. Using the new scheme, high accuracy is achieved at a low computational cost. Salient features of the new approach are highlighted by a numerical example.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78531150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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