Consider the following 2×2 nonlinear system:where f(u): R→R is a, smooth function. Setwhere F’(u)= f(u). Then (1) can be rewritten as an equivalent Hamiltonian system:
This paper presents both analytical and numerical studies of the conservative Sawada-Kotera equation and its dissipative generalization,equations known for their soliton solutions and rich chaotic dynamics.These model...This paper presents both analytical and numerical studies of the conservative Sawada-Kotera equation and its dissipative generalization,equations known for their soliton solutions and rich chaotic dynamics.These models offer valuable insights into nonlinear wave propagation,with applications in fluid dynamics and materials science,including systems such as liquid crystals and ferrofluids.It is shown that the conservative Sawada-Kotera equation supports traveling wave solutions corresponding to elliptic limit cycles,as well as two-and three-dimensional invariant tori surrounding these cycles in the associated ordinary differential equation(ODE)system.For the dissipative generalized Sawada-Kotera equation,chaotic wave behavior is observed.The transition to chaos in the corresponding ODE systemfollows a universal bifurcation scenario consistent with the framework established by FShM(Feigenbaum-Sharkovsky-Magnitskii)theory.Notably,this study demonstrates for the first time that the conservative Sawada-Kotera equation can exhibit complex quasi-periodic wave solutions,while its dissipative counterpart admits an infinite number of stable periodic and chaotic waveforms.展开更多
A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CW...A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CWENO） reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.展开更多
In this paper we study the problem of the global existence （in time） of weak entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models th...In this paper we study the problem of the global existence （in time） of weak entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit （large） threshold, then the Cauchy problem has global solutions.展开更多
We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form.It applies in multidimensional structured and unstructured meshes.The proposed method is ...We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form.It applies in multidimensional structured and unstructured meshes.The proposed method is an extension of the UFORCEmethod developed by Stecca,Siviglia and Toro[25],in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan.The proposed first-order method is shown to be identical to the Godunov upwindmethod in applications to a 2×2 linear hyperbolic system.The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations.Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes.Finally,numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.展开更多
This paper analyzes the dynamic behavior of a two-degree-of-freedom system subjected to electromagnetic interaction modelled through a skew-symmetric coupling matrix.The system comprises two mechanically independent o...This paper analyzes the dynamic behavior of a two-degree-of-freedom system subjected to electromagnetic interaction modelled through a skew-symmetric coupling matrix.The system comprises two mechanically independent oscillators coupled by velocity-dependent electromagnetic forces.The equations of motion are formulated and analyzed in the modal domain,highlighting the effects of the antisymmetric interaction on natural frequencies and mode shapes.The classical orthogonality is broken,resulting in complex eigenvectors;nevertheless,the system remains conservative,as the interaction forces perform no work.The analysis is carried out using both configuration-space and state-space formulations,revealing modal frequency splitting and phase shifts induced by the skew-symmetric term.These modal features are further examined through time-domain simulations and frequency response functions.The main contribution of this study is the development and analysis of a deliberately simple yet general model that isolates the essential dynamic effects of skew-symmetric electromagnetic coupling.This minimal formulation,often hidden in more complex systems,reveals key phenomena such as modal frequency splitting,non-normal modes,and energy-conserving cross-effects.The model serves not only as a conceptual reference but also as a methodological framework applicable to a broad class of coupled electromechanical systems.展开更多
The numerical simulation of non conservative system is a difficult challenge for two reasons at least.The first one is that it is not possible to derive jump relations directly from conservation principles,so that in ...The numerical simulation of non conservative system is a difficult challenge for two reasons at least.The first one is that it is not possible to derive jump relations directly from conservation principles,so that in general,if the model description is non ambiguous for smooth solutions,this is no longer the case for discontinuous solutions.From the numerical view point,this leads to the following situation:if a scheme is stable,its limit for mesh convergence will depend on its dissipative structure.This is well known since at least[1].In this paper we are interested in the“dual”problem:given a system in non conservative form and consistent jump relations,how can we construct a numerical scheme that will,for mesh convergence,provide limit solutions that are the exact solution of the problem.In order to investigate this problem,we consider a multiphase flow model for which jump relations are known.Our scheme is an hybridation of Glimm scheme and Roe scheme.展开更多
We study central-upwind schemes for systems of hyperbolic conservation laws,recently introduced in[13].Similarly to staggered non-oscillatory central schemes,these schemes are central Godunov-type projection-evolution...We study central-upwind schemes for systems of hyperbolic conservation laws,recently introduced in[13].Similarly to staggered non-oscillatory central schemes,these schemes are central Godunov-type projection-evolution methods that enjoy the advantages of high resolution,simplicity,universality and robustness.At the same time,the central-upwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes.In this paper,we present a modification of the one-dimensional fully-and semi-discrete central-upwind schemes,in which the numerical dissipation is reduced even further.The goal is achieved by a more accurate projection of the evolved quantities onto the original grid.In the semi-discrete case,the reduction of dissipation procedure leads to a new,less dissipative numerical flux.We also extend the new semi-discrete scheme to the twodimensional case via the rigorous,genuinely multidimensional derivation.The new semi-discrete schemes are tested on a number of numerical examples,where one can observe an improved resolution,especially of the contact waves.展开更多
This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation.The prototype,extended from a 1D model,reduces substantially less dissipation than expec...This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation.The prototype,extended from a 1D model,reduces substantially less dissipation than expected.The problem arises from over-restriction of some slope limiters,which keep slopes between interfaces of cells to be Total-Variation-Diminishing.This study reports the defect and presents a re-derived optimal formula.Numerical experiments highlight the significance of this formula,especially in long-time,large-scale simulations.展开更多
基金Supported by NSFC (Grant No. 10071030) partially by Volkswagen Stiftung, Germany
文摘Consider the following 2×2 nonlinear system:where f(u): R→R is a, smooth function. Setwhere F’(u)= f(u). Then (1) can be rewritten as an equivalent Hamiltonian system:
文摘This paper presents both analytical and numerical studies of the conservative Sawada-Kotera equation and its dissipative generalization,equations known for their soliton solutions and rich chaotic dynamics.These models offer valuable insights into nonlinear wave propagation,with applications in fluid dynamics and materials science,including systems such as liquid crystals and ferrofluids.It is shown that the conservative Sawada-Kotera equation supports traveling wave solutions corresponding to elliptic limit cycles,as well as two-and three-dimensional invariant tori surrounding these cycles in the associated ordinary differential equation(ODE)system.For the dissipative generalized Sawada-Kotera equation,chaotic wave behavior is observed.The transition to chaos in the corresponding ODE systemfollows a universal bifurcation scenario consistent with the framework established by FShM(Feigenbaum-Sharkovsky-Magnitskii)theory.Notably,this study demonstrates for the first time that the conservative Sawada-Kotera equation can exhibit complex quasi-periodic wave solutions,while its dissipative counterpart admits an infinite number of stable periodic and chaotic waveforms.
基金the National Natural Science Foundation of China (60134010)The English text was polished by Yunming Chen.
文摘A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CWENO） reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.
文摘In this paper we study the problem of the global existence （in time） of weak entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit （large） threshold, then the Cauchy problem has global solutions.
文摘We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form.It applies in multidimensional structured and unstructured meshes.The proposed method is an extension of the UFORCEmethod developed by Stecca,Siviglia and Toro[25],in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan.The proposed first-order method is shown to be identical to the Godunov upwindmethod in applications to a 2×2 linear hyperbolic system.The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations.Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes.Finally,numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.
基金supported by the Department of Education of the Basque Government for the Research Group program IT1507–22.
文摘This paper analyzes the dynamic behavior of a two-degree-of-freedom system subjected to electromagnetic interaction modelled through a skew-symmetric coupling matrix.The system comprises two mechanically independent oscillators coupled by velocity-dependent electromagnetic forces.The equations of motion are formulated and analyzed in the modal domain,highlighting the effects of the antisymmetric interaction on natural frequencies and mode shapes.The classical orthogonality is broken,resulting in complex eigenvectors;nevertheless,the system remains conservative,as the interaction forces perform no work.The analysis is carried out using both configuration-space and state-space formulations,revealing modal frequency splitting and phase shifts induced by the skew-symmetric term.These modal features are further examined through time-domain simulations and frequency response functions.The main contribution of this study is the development and analysis of a deliberately simple yet general model that isolates the essential dynamic effects of skew-symmetric electromagnetic coupling.This minimal formulation,often hidden in more complex systems,reveals key phenomena such as modal frequency splitting,non-normal modes,and energy-conserving cross-effects.The model serves not only as a conceptual reference but also as a methodological framework applicable to a broad class of coupled electromechanical systems.
基金funded in part by the EU ERC Advanced grant“ADDECCO”#226616This work has been done in part while H.Kumar was a post doc at INRIA,funded by the EU ERC Advanced grant“ADDECCO”#226616.
文摘The numerical simulation of non conservative system is a difficult challenge for two reasons at least.The first one is that it is not possible to derive jump relations directly from conservation principles,so that in general,if the model description is non ambiguous for smooth solutions,this is no longer the case for discontinuous solutions.From the numerical view point,this leads to the following situation:if a scheme is stable,its limit for mesh convergence will depend on its dissipative structure.This is well known since at least[1].In this paper we are interested in the“dual”problem:given a system in non conservative form and consistent jump relations,how can we construct a numerical scheme that will,for mesh convergence,provide limit solutions that are the exact solution of the problem.In order to investigate this problem,we consider a multiphase flow model for which jump relations are known.Our scheme is an hybridation of Glimm scheme and Roe scheme.
基金supported in part by the NSF Grant DMS-0310585The work of C.-T.Lin was supported in part by the NSC grants NSC 94-2115-M-126-003 and 91-2115-M-126-001.
文摘We study central-upwind schemes for systems of hyperbolic conservation laws,recently introduced in[13].Similarly to staggered non-oscillatory central schemes,these schemes are central Godunov-type projection-evolution methods that enjoy the advantages of high resolution,simplicity,universality and robustness.At the same time,the central-upwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes.In this paper,we present a modification of the one-dimensional fully-and semi-discrete central-upwind schemes,in which the numerical dissipation is reduced even further.The goal is achieved by a more accurate projection of the evolved quantities onto the original grid.In the semi-discrete case,the reduction of dissipation procedure leads to a new,less dissipative numerical flux.We also extend the new semi-discrete scheme to the twodimensional case via the rigorous,genuinely multidimensional derivation.The new semi-discrete schemes are tested on a number of numerical examples,where one can observe an improved resolution,especially of the contact waves.
文摘This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation.The prototype,extended from a 1D model,reduces substantially less dissipation than expected.The problem arises from over-restriction of some slope limiters,which keep slopes between interfaces of cells to be Total-Variation-Diminishing.This study reports the defect and presents a re-derived optimal formula.Numerical experiments highlight the significance of this formula,especially in long-time,large-scale simulations.
