Variational Integrators for Reduced Magnetohydrodynamics

Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics equations with applications to both fusion and astrophysical plasmas, possessing a noncanonical Hamiltonian structure and consequently a number of conserved functionals. We propose a new discretisation strategy for these equations based on a discrete variational principle applied to a formal Lagrangian. The resulting integrator preserves important quantities like the total energy, magnetic helicity and cross helicity exactly (up to machine precision). As the integrator is free of numerical resistivity, spurious reconnection along current sheets is absent in the ideal case. If effects of electron inertia are added, reconnection of magnetic field lines is allowed, although the resulting model still possesses a noncanonical Hamiltonian structure. After reviewing the conservation laws of the model equations, the adopted variational principle with the related conservation laws are described both at the continuous and discrete level. We verify the favourable properties of the variational integrator in particular with respect to the preservation of the invariants of the models under consideration and compare with results from the literature and those of a pseudo-spectral code.Comment: 35 page

C1-continuous space-time discretization based on Hamilton's law of varying action

We develop a class of C1-continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1-continuity for the displacement field and C0-continuity for the velocity field. From the discrete virtual action we finally construct a class of one-step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one-dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate -- denoted as p2-scheme -- converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long-term energy behavior.Comment: slightly condensed the manuscript, added references, numerical results unchange

Polynomial mechanics and optimal control

We describe a new algorithm for trajectory optimization of mechanical systems. Our method combines pseudo-spectral methods for function approximation with variational discretization schemes that exactly preserve conserved mechanical quantities such as momentum. We thus obtain a global discretization of the Lagrange-d'Alembert variational principle using pseudo-spectral methods. Our proposed scheme inherits the numerical convergence characteristics of spectral methods, yet preserves momentum-conservation and symplecticity after discretization. We compare this algorithm against two other established methods for two examples of underactuated mechanical systems; minimum-effort swing-up of a two-link and a three-link acrobot.Comment: Final version to EC