Hamiltonian systems are canonical systems on phase space endowed with symplectic structures. The dynamical evolutions, i.e., the phase flow of the Hamiltonian systems are symplectic transformations which are area-pres...
Hamiltonian systems are canonical systems on phase space endowed with symplectic structures. The dynamical evolutions, i.e., the phase flow of the Hamiltonian systems are symplectic transformations which are area-preserving. The importance of the Hamiltonian systems and their special property require the numerical algorithms for them should preserve as much as possible the relevant symplectic properties of the original systems. Feng Kang [1-3] proposed in 1984 a new approach to computing Hamiltonian systems from the view point of symplectic geometry. He systematically described the general method for constructing symplectic schemes with any order accuracy via generating functions. A generalization of the above theory and methods for canonical Hamiltonian equations in infinite dimension can be found in [4]. Using self-adjoint schemes, we can construct schemes of arbitrary even order [5]. These schemes can be applied to wave equation [6,7] and the stability of them can be seen in [7,8]. In this paper, we will use the hyperbolic functions sinh(x), cosh(x) and tanh(x) to construct symplectic schemes of arbitrary order for wave equations and stabilities of these constructed schemes are also discussed.
Marine foundation species are the biotic basis for many of the world's coastal ecosystems, providing structural habitat, food, and protection for myriad plants and animals as well as many ecosystem services. ...
Marine foundation species are the biotic basis for many of the world's coastal ecosystems, providing structural habitat, food, and protection for myriad plants and animals as well as many ecosystem services. However, climate change poses a significant threat to foundation species and the ecosystems they support. We review the impacts of climate change on common marine foundation species, including corals, kelps, seagrasses, salt marsh plants, mangroves, and bivalves. It is evident that marine foundation species have already been severely impacted by several climate change drivers, often through interactive effects with other human stressors, such as pollution, overfishing, and coastal development. Despite considerable variation in geographical, environmental, and ecological contexts, direct and indirect effects of gradual warming and subsequent heatwaves have emerged as the most pervasive drivers of observed impact and potent threat across all marine foundation species, but effects from sea level rise, ocean acidification, and increased storminess are expected to increase. Documented impacts include changes in the genetic structures, physiology, abundance, and distribution of the foundation species themselves and changes to their interactions with other species, with flow-on effects to associated communities, biodiversity, and ecosystem functioning. We discuss strategies to support marine foundation species into the Anthropocene, in order to increase their resilience and ensure the persistence of the ecosystem services they provide.
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