Explicit local time-stepping methods for time-dependent wave propagation

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leap-frog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations validate the theory and illustrate the usefulness of these local time-stepping methods.Comment: overview paper, typos added, references updated. arXiv admin note: substantial text overlap with arXiv:1109.448

ParaExp using Leapfrog as Integrator for High-Frequency Electromagnetic Simulations

Recently, ParaExp was proposed for the time integration of linear hyperbolic problems. It splits the time interval of interest into sub-intervals and computes the solution on each sub-interval in parallel. The overall solution is decomposed into a particular solution defined on each sub-interval with zero initial conditions and a homogeneous solution propagated by the matrix exponential applied to the initial conditions. The efficiency of the method depends on fast approximations of this matrix exponential based on recent results from numerical linear algebra. This paper deals with the application of ParaExp in combination with Leapfrog to electromagnetic wave problems in time-domain. Numerical tests are carried out for a simple toy problem and a realistic spiral inductor model discretized by the Finite Integration Technique.Comment: Corrected typos. arXiv admin note: text overlap with arXiv:1607.0036

Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control

This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating the fractional problem either by discretizing the fractional term or expanding the fractional derivatives as a series involving integer order derivatives. The former method, as a subclass of direct methods in the theory of calculus of variations, uses finite differences, Grunwald-Letnikov definition in this case, to discretize the fractional term. Any quadrature rule for integration, regarding the desired accuracy, is then used to discretize the whole problem including constraints. The final task in this method is to solve a static optimization problem to reach approximated values of the unknown functions on some mesh points. The latter method, however, approximates fractional problems by classical ones in which only derivatives of integer order are present. Precisely, two continuous approximations for fractional derivatives by series involving ordinary derivatives are introduced. Local upper bounds for truncation errors are provided and, through some test functions, the accuracy of the approximations are justified. Then we substitute the fractional term in the original problem with these series and transform the fractional problem to an ordinary one. Hereafter, we use indirect methods of classical theory, e.g. Euler-Lagrange equations, to solve the approximated problem. The methods are mainly developed through some concrete examples which either have obvious solutions or the solution is computed using the fractional Euler-Lagrange equation.Comment: This is a preprint of a paper whose final and definite form appeared in: Chapter V, Fractional Calculus in Analysis, Dynamics and Optimal Control (Editor: Jacky Cresson), Series: Mathematics Research Developments, Nova Science Publishers, New York, 2014. (See http://www.novapublishers.com/catalog/product_info.php?products_id=46851). Consists of 39 page

Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems

Explicit Runge-Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretization on quadrilateral grids. The new schemes permit an effective time step that is substantially larger than the maximum admissible time step of standard explicit Runge-Kutta schemes available in literature. Furthermore, they have a small principal error norm and admit a low-storage implementation. The advantages of the new schemes are demonstrated through application to the Euler equations and the linearized Euler equations.Comment: 37 pages, 3 pages of appendi