Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity. In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and coworkers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.
{"title":"Locally conservative finite difference schemes for the modified KdV equation","authors":"Gianluca Frasca-Caccia, P. Hydon","doi":"10.3934/jcd.2019015","DOIUrl":"https://doi.org/10.3934/jcd.2019015","url":null,"abstract":"Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity. In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and coworkers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83225892","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}
In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each $k,ninmathbb N$ with $ngeqslant 2k+1$ we obtain a Lotka-Volterra system $hbox{LV}_b(n,k)$ on $mathbb R^n$ which is a deformation of the Lotka-Volterra system $hbox{LV}(n,k)$, which is itself an integrable reduction of the $m$-dimensional Bogoyavlenskij-Itoh system $hbox{LV}(2m+1,m)$, where $m=n-k-1$. We prove that $hbox{LV}_b(n,k)$ is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational integrals of $hbox{LV}(n,k)$. We also construct a family of discretizations of $hbox{LV}_b(n,0)$, including its Kahan discretization, and we show that these discretizations are also Liouville and superintegrable.
{"title":"Integrable reductions of the dressing chain","authors":"C. Evripidou, P. Kassotakis, P. Vanhaecke","doi":"10.3934/jcd.2019014","DOIUrl":"https://doi.org/10.3934/jcd.2019014","url":null,"abstract":"In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each $k,ninmathbb N$ with $ngeqslant 2k+1$ we obtain a Lotka-Volterra system $hbox{LV}_b(n,k)$ on $mathbb R^n$ which is a deformation of the Lotka-Volterra system $hbox{LV}(n,k)$, which is itself an integrable reduction of the $m$-dimensional Bogoyavlenskij-Itoh system $hbox{LV}(2m+1,m)$, where $m=n-k-1$. We prove that $hbox{LV}_b(n,k)$ is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational integrals of $hbox{LV}(n,k)$. We also construct a family of discretizations of $hbox{LV}_b(n,0)$, including its Kahan discretization, and we show that these discretizations are also Liouville and superintegrable.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89971077","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}
A parameter-dependent class of Hamiltonian (generalized) Lotka-Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.
{"title":"A new class of integrable Lotka–Volterra systems","authors":"H. Christodoulidi, A. Hone, T. Kouloukas","doi":"10.3934/jcd.2019011","DOIUrl":"https://doi.org/10.3934/jcd.2019011","url":null,"abstract":"A parameter-dependent class of Hamiltonian (generalized) Lotka-Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86838825","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}
In this paper, we exploit the gradient flow structure of continuous-time formulations of Bayesian inference in terms of their numerical time-stepping. We focus on two particular examples, namely, the continuous-time ensemble Kalman-Bucy filter and a particle discretisation of the Fokker-Planck equation associated to Brownian dynamics. Both formulations can lead to stiff differential equations which require special numerical methods for their efficient numerical implementation. We compare discrete gradient methods to alternative semi-implicit and other iterative implementations of the underlying Bayesian inference problems.
{"title":"Discrete gradients for computational Bayesian inference","authors":"S. Pathiraja, S. Reich","doi":"10.3934/jcd.2019019","DOIUrl":"https://doi.org/10.3934/jcd.2019019","url":null,"abstract":"In this paper, we exploit the gradient flow structure of continuous-time formulations of Bayesian inference in terms of their numerical time-stepping. We focus on two particular examples, namely, the continuous-time ensemble Kalman-Bucy filter and a particle discretisation of the Fokker-Planck equation associated to Brownian dynamics. Both formulations can lead to stiff differential equations which require special numerical methods for their efficient numerical implementation. We compare discrete gradient methods to alternative semi-implicit and other iterative implementations of the underlying Bayesian inference problems.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77375688","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}
Y. Miyatake, T. Nakagawa, T. Sogabe, Shaoliang Zhang
We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schr"odinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.
{"title":"A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation","authors":"Y. Miyatake, T. Nakagawa, T. Sogabe, Shaoliang Zhang","doi":"10.3934/jcd.2019018","DOIUrl":"https://doi.org/10.3934/jcd.2019018","url":null,"abstract":"We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schr\"odinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86857367","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}
{"title":"Linear degree growth in lattice equations","authors":"Dinh T Tran, John A. G. Roberts","doi":"10.3934/jcd.2019023","DOIUrl":"https://doi.org/10.3934/jcd.2019023","url":null,"abstract":"","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74475530","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}
In plasma simulations, numerical methods with high computational efficiency and long-term stability are needed. In this paper, symplectic methods with adaptive time steps are constructed for simulating the dynamics of charged particles under the electromagnetic field. With specifically designed step size functions, the motion of charged particles confined in a Penning trap under three different magnetic fields is studied, and also the dynamics of runaway electrons in tokamaks is investigated. The numerical experiments are performed to show the efficiency of the new derived adaptive symplectic methods.
{"title":"Study of adaptive symplectic methods for simulating charged particle dynamics","authors":"Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu","doi":"10.3934/jcd.2019022","DOIUrl":"https://doi.org/10.3934/jcd.2019022","url":null,"abstract":"In plasma simulations, numerical methods with high computational efficiency and long-term stability are needed. In this paper, symplectic methods with adaptive time steps are constructed for simulating the dynamics of charged particles under the electromagnetic field. With specifically designed step size functions, the motion of charged particles confined in a Penning trap under three different magnetic fields is studied, and also the dynamics of runaway electrons in tokamaks is investigated. The numerical experiments are performed to show the efficiency of the new derived adaptive symplectic methods.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75521042","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}
Reinout Quispel was born on 8 October 1953 in Bilthoven, a small town near Utrecht in the Netherlands. He studied both chemistry and physics, gaining bachelor’s degrees at the University of Utrecht in 1973 and 1976 respectively, and then specialized in theoretical physics, with a Master’s degree in 1979 (on solitons in the Heisenberg spin chain, supervised by Theodorus Ruijgrok) and a PhD, Linear Integral Equations and Soliton Systems [22], in 1983, supervised by Hans Capel. This thesis, which begins with a study of integrable PDEs, arrives in Chapter 4 (later published in [24]) with the discovery of a method for obtaining fully discrete integrable systems on square lattices, that have as continuum limits the Korteweg–de Vries, nonlinear Schrödinger, and complex sine–Gordon equations, and the Heisenberg spin chain. Thus several of Reinout’s lifelong research interests – continuous and discrete integrability, and the relationship between the continuous and the discrete – were present right from the start. The next stop was a postdoc at Twente University, working with Robert Helleman, the founder of the ‘Dynamics Days’ conference series, before a long-distance move to the Australian National University, working with Rodney Baxter. Reinout and Nel expected this southern sojourn to last for three years; thirty-three years later they are still happily resident in Australia. In 1990 Reinout moved to La Trobe University, Melbourne, where he became a Professor in 2004. Reinout’s three main research areas are discrete integrable systems, dynamical systems, and geometric numerical integration, along with interactions between these topics. In discrete integrable systems, having introduced a major new direction in his PhD thesis – his novel reductions to Painlevé equations led to the Clarkson–Kruskal non-classical reduction method – he continued by codiscovering the QRT map [25, 26], an 18-parameter family of completely integrable maps of the plane. These turned out to have far-reaching implications in dynamical systems theory, geometry, and integrability. For example, the modern construction of nonautonomous dynamical systems known as discrete Painlevé equations rely on them. Their geometry is explored at length in the 2010 book QRT and Elliptic Surfaces by Hans Duistermaat and is still being investigated today. In dynamical systems, his work has centred on systems with discrete and/or continuous symmetries. His review [28] marked the emergence of reversible dynamical
{"title":"Preface Special issue in honor of Reinout Quispel","authors":"E. Celledoni, R. McLachlan","doi":"10.3934/jcd.2019007","DOIUrl":"https://doi.org/10.3934/jcd.2019007","url":null,"abstract":"Reinout Quispel was born on 8 October 1953 in Bilthoven, a small town near Utrecht in the Netherlands. He studied both chemistry and physics, gaining bachelor’s degrees at the University of Utrecht in 1973 and 1976 respectively, and then specialized in theoretical physics, with a Master’s degree in 1979 (on solitons in the Heisenberg spin chain, supervised by Theodorus Ruijgrok) and a PhD, Linear Integral Equations and Soliton Systems [22], in 1983, supervised by Hans Capel. This thesis, which begins with a study of integrable PDEs, arrives in Chapter 4 (later published in [24]) with the discovery of a method for obtaining fully discrete integrable systems on square lattices, that have as continuum limits the Korteweg–de Vries, nonlinear Schrödinger, and complex sine–Gordon equations, and the Heisenberg spin chain. Thus several of Reinout’s lifelong research interests – continuous and discrete integrability, and the relationship between the continuous and the discrete – were present right from the start. The next stop was a postdoc at Twente University, working with Robert Helleman, the founder of the ‘Dynamics Days’ conference series, before a long-distance move to the Australian National University, working with Rodney Baxter. Reinout and Nel expected this southern sojourn to last for three years; thirty-three years later they are still happily resident in Australia. In 1990 Reinout moved to La Trobe University, Melbourne, where he became a Professor in 2004. Reinout’s three main research areas are discrete integrable systems, dynamical systems, and geometric numerical integration, along with interactions between these topics. In discrete integrable systems, having introduced a major new direction in his PhD thesis – his novel reductions to Painlevé equations led to the Clarkson–Kruskal non-classical reduction method – he continued by codiscovering the QRT map [25, 26], an 18-parameter family of completely integrable maps of the plane. These turned out to have far-reaching implications in dynamical systems theory, geometry, and integrability. For example, the modern construction of nonautonomous dynamical systems known as discrete Painlevé equations rely on them. Their geometry is explored at length in the 2010 book QRT and Elliptic Surfaces by Hans Duistermaat and is still being investigated today. In dynamical systems, his work has centred on systems with discrete and/or continuous symmetries. His review [28] marked the emergence of reversible dynamical","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88786807","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}
A system of ordinary differential equations of a predator–prey type, depending on nine parameters, is studied. We have included in this model a nonmonotonic response function and time periodic perturbation. Using numerical continuation software, we have detected three codimension two bifurcations for the unperturbed system, namely cusp, Bogdanov-Takens and Bautin bifurcations. Furthermore, we concentrate on two regions in the parameter space, the region where the Bogdanov-Takens and the region where Bautin bifurcations occur. As we turn on the time perturbation, we find strange attractors in the neighborhood of invariant tori of the unperturbed system.
{"title":"Strange attractors in a predator–prey system with non-monotonic response function and periodic perturbation","authors":"J. M. Tuwankotta, Eric Harjanto","doi":"10.3934/jcd.2019024","DOIUrl":"https://doi.org/10.3934/jcd.2019024","url":null,"abstract":"A system of ordinary differential equations of a predator–prey type, depending on nine parameters, is studied. We have included in this model a nonmonotonic response function and time periodic perturbation. Using numerical continuation software, we have detected three codimension two bifurcations for the unperturbed system, namely cusp, Bogdanov-Takens and Bautin bifurcations. Furthermore, we concentrate on two regions in the parameter space, the region where the Bogdanov-Takens and the region where Bautin bifurcations occur. As we turn on the time perturbation, we find strange attractors in the neighborhood of invariant tori of the unperturbed system.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79608247","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 dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [ 18 ]. In this framework, the dynamics of the begin{document}$ j $end{document} th body is described in a frame relative to the begin{document}$ (j-1) $end{document} th one. Starting from the Lagrangian formulation of the system on begin{document}$ {{rm{SO}}}(3)^{N} $end{document} , the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.
We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [ 18 ]. In this framework, the dynamics of the begin{document}$ j $end{document} th body is described in a frame relative to the begin{document}$ (j-1) $end{document} th one. Starting from the Lagrangian formulation of the system on begin{document}$ {{rm{SO}}}(3)^{N} $end{document} , the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.
{"title":"Chains of rigid bodies and their numerical simulation by local frame methods","authors":"N. Sætran, A. Zanna","doi":"10.3934/jcd.2019021","DOIUrl":"https://doi.org/10.3934/jcd.2019021","url":null,"abstract":"We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [ 18 ]. In this framework, the dynamics of the begin{document}$ j $end{document} th body is described in a frame relative to the begin{document}$ (j-1) $end{document} th one. Starting from the Lagrangian formulation of the system on begin{document}$ {{rm{SO}}}(3)^{N} $end{document} , the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82007943","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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