Generalized Galerkin Variational Integrators
Generalized Galerkin Variational Integrators
复制标题
广义伽辽金变分积分器
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
M. Leok
中科院分区：
文献类型：
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作者：
M. Leok
我们引入广义伽辽金变分积分器，它是离散变分力学的自然推广，其中离散作用，而不是离散拉格朗日，是基本对象。这是通过适当选择有限维函数空间来近似作用积分来实现的，该有限维函数空间近似配置束的部分和数值求积来近似积分。我们讨论这个通用框架如何使我们能够恢复高阶伽辽金变分积分器、异步变分积分器和辛能量动量积分器。此外，我们将考虑未由节点处计算的场值参数化的函数空间，这允许构造李群、多尺度和伪谱变分积分器。通过将伪谱变分积分器应用于（线性）薛定谔方程来说明其构造。 G 不变离散拉格朗日量是在李群方法的背景下通过使用自然图表和李代数级别的插值来构造的。这些 G 不变拉格朗日量的约简产生了离散欧拉-庞加莱约简的高阶类似物。通过考虑非线性逼近空间，还可以引入时空自适应变分积分器。
We introduce generalized Galerkin variational integrators, which are a natural generalization of discrete variational mechanics, whereby the discrete action, as opposed to the discrete Lagrangian, is the fundamental object. This is achieved by approximating the action integral with appropriate choices of a finite-dimensional function space that approximate sections of the configuration bundle and numerical quadrature to approximate the integral. We discuss how this general framework allows us to recover higher-order Galerkin variational integrators, asynchronous variational integrators, and symplectic-energy-momentum integrators. In addition, we will consider function spaces that are not parameterized by field values evaluated at nodal points, which allows the construction of Lie group, multiscale, and pseudospectral variational integrators. The construction of pseudospectral variational integrators is illustrated by applying it to the (linear) Schrodinger equation. G-invariant discrete Lagrangians are constructed in the context of Lie group methods through the use of natural charts and interpolation at the level of the Lie algebra. The reduction of these G-invariant Lagrangians yield a higher-order analogue of discrete Euler-Poincare reduction. By considering nonlinear approximation spaces, spatio-temporally adaptive variational integrators can be introduced as well.