Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations

The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the Störmer-Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also include

Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review

Numerical methods for oscillatory, multi-scale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time- or state-dependent frequencies is emphasized. Trigonometric integrators for the first case and adiabatic integrators for the second case are discussed in more detail

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

The final publication is available at Springer via http://dx.doi.org/10.1007/s10915-016-0319-1In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(τ^(3−α) +h^2 ),

Extrapolation integrators for constrained multibody systems

Extrapolation methods using the structure in the equations of motion of multibody systems are given in this article. The methods are explicit in the differential part and implicit in the nonlinear constraints. They admit a robust formulation in which only linear systems of equations are solved most of the time. Related methods, which are linearly implicit also in the differential part, are developed for stiff mechanical systems. Numerical results for the extrapolation code MEXX are included. 0 1991 Academic PXS. hc. 1

On the Multistep Time Discretization of Linear Initial-Boundary Value Problems and Their Boundary Integral Equations

Introduction The present article is about time discretization methods for linear time-invariant non-homogeneous evolution equations. These include initial-boundary value problems for partial differential equations of hyperbolic and parabolic type, and boundary integral equations for such problems. A common feature is that the solution operator is a temporal convolution k g with the data g. Here, the (distributional) convolution kernel k is not known explicitly, but various concepts of well-posedness can be phrased in terms of bounds for its Laplace transform K(s), for s varying in a half-plane Re s ? oe 0 . Even K(s), which is the solution operator of the Laplace transformed problem, is usually not known, but it is modeled implicitly in the time discretization of partial differential equations by linear multis