We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on previous work on time-dependent discrete mechanics, our approach is based on a discrete version of the Lagrange-d'Alembert principle for nonautonomous systems.
We propose a numerical scheme for the time-integration of nonholonomic mechanical systems, both conservative and nonconservative. The scheme is obtained by simultaneously discretizing the constraint equations and the Herglotz variational principle. We validate the method using numerical simulations and contrast them against the results of standard methods from the literature.
In this paper, high-order numerical integrators on homogeneous spaces will be presented as an application of nonholonomic partitioned Runge-Kutta Munthe-Kaas (RKMK) methods on Lie groups. A homogeneous space $M$ is a manifold where a group $G$ acts transitively. Such a space can be understood as a quotient $M \cong G/H$, where $H$ a closed Lie subgroup, is the isotropy group of each point of $M$. The Lie algebra of $G$ may be decomposed into $\mathfrak{g} = \mathfrak{m} \oplus \mathfrak{h}$, where $\mathfrak{h}$ is the subalgebra that generates $H$ and $\mathfrak{m}$ is a subspace. Thus, varia…
New developments on the Geometric Nonholonomic Integrator
Universidad Nacional del Sur（南方大学）；Instituto de Matemática Bahía Blanca（巴塔哥尼亚数学研究所）；CONICET（阿根廷国家科学研究委员会）；Zentrum Mathematik der Technische Universität München（慕尼黑工业大学数学中心）；Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM（马德里高等科学研究院（由CSIC、UAM、UC3M、UCM联合组建））
We construct the exponential map associated to a nonholonomic system that allows us to define an exact discrete nonholonomic constraint submanifold. We reproduce the continuous nonholonomic flow as a discrete flow on this discrete constraint submanifold deriving an exact discrete version of the nonholonomic equations. Finally, we derive a general family of nonholonomic integrators.
In this paper, we develop a framework to construct energy-preserving methods for multi-components Hamiltonian systems, combining the exponential integrator and the partitioned averaged vector field method. This leads to numerical schemes with both advantages of long-time stability and excellent behavior for highly oscillatory or stiff problems. Compared to the existing energy-preserving exponential integrators (EP-EI) in practical implementation, our proposed methods are much efficient which can at least be computed by subsystem instead of handling a nonlinear coupling system at a time. Moreov…
A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schrödinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation is first reformulated into an equivalent form which satisfies a quadratic energy. The spatial derivatives of the system are then approximated with the standard Fourier pseudo-spectral method. Subsequently, we apply the extrapolation technique/prediction-correction strategy to the nonlinear terms of the semi-discretized system and a linearized energy-conservin…
We study the conservation properties of the Hermite-discontinuous Galerkin (Hermite-DG) approximation of the Vlasov-Maxwell equations. In this semi-discrete formulation, the total mass is preserved independently for every plasma species. Further, an energy invariant exists if central numerical fluxes are used in the DG approximation of Maxwell's equations, while a dissipative term is present when upwind fluxes are employed. In general, traditional temporal integrators might fail to preserve invariants associated with conservation laws during the time evolution. Hence, we analyze the capability…
Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods
Dipartimento di Matematica e Informatica “U. Dini”, Viale Morgagni 67/a, 50134 Firenze, Italy（都尼数学与信息系）；Dipartimento di Matematica, Via Orabona 4, 70125 Bari, Italy（巴里大学数学系）
A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models
Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy, but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. …
Energy methods for constructing time-stepping algorithms are of increased interest in application to nonlinear problems, since numerical stability can be inferred from the conservation of the system energy. Alternatively, symplectic integrators may be constructed that preserve the symplectic form of the system. This methodology has been established for Hamiltonian systems, with numerous applications in engineering problems. In this paper an extension of such methods to non-conservative acoustic systems is presented. Discrete conservation laws, equivalent to that of energy-conserving schemes, a…
