1st Canadian Symposium in Numerical Analysis and Scientific Computing
Monday, June 17th
Salle Des Plaines B
Adaptive space-time finite element method for a nonlinear form of Maxwell's equations in applied superconductivity
Laforest, Marc
École Polytechnique de Montréal

The steady improvement of the performances of high temperature superconductors (HTS) brings them within reach of new applications, such as HTS motors, transformers and fault current limiters. To optimize the design of these devices, one must be able to predict the magnetic and electric fields in complex 3-D geometries, but doing so efficiently and accurately is still a challenging task. Within the engineering community, phenomenological models relating the electric field and current density of HTS lead to a novel nonlinear evolutionary monotone PDE based on Maxwell's equations, which is effectively a generalization of the classical p-Laplacian problem. Unfortunately, these models possesses sharp moving fronts that lead to the use of prohibitively small time steps in numerical simulations, even in 2-D domains.

In this work, we propose a new numerical space-time method that allows for local space and time adaptivity without the restrictive global timestep constraint. We present an a posteriori error estimator for the computation of the AC loss, a key design parameter for HTS devices. Numerical results are presented in one and two space dimensions attesting to the efficiency of the numerical method. This work was done in collaboration with Andy Wan and Frédéric Sirois, both from the École Polytechnique de Montréal.

High-Order Energy-Conserved S-FDTD Schemes for Computational Electromagnetics
Liang, Dong
Department of Mathematics and Statistics, York University

In computational electromagnetics, for problems requiring long-time integration and problems of wave propagations over longer distances, it has led to the development of high-order FDTD schemes which produce smaller dispersion or phase errors for a given mesh resolution. However, some delevoped high-order FDTD schemes are conditionally stable and require large computational memory and huge computational cost. On the other hand, during the propagation of electromagnetic waves in lossless media without sources, the electromagnetic energy keeps constant for all time, which explains the physical feature of conservation of electromagnetic energy in long term behavior. It is significantly important to preserve this invariance in time. Thus, developing high-order energy-conserved S-FDTD schemes, for Maxwell's equations and specially for a long term computation of electromagnetic fields, is very important and challenging, which will provide to satisfy discrete energy conservations, unconditional stability, non-dissipativity, and high-order accuracy. In this talk, we will present our new high-order energy-conserved S-FDTD schemes for Maxwell's equations. We will show theoretical results on energy conservation, unconditional stability and optimal convergence. We will also present numerical experiments to confirm our theoretical results.

A conservative adaptive wavelet method for the shallow water equations on staggered grids
Kevlahan, Nicholas
McMaster University

This talk presents the first dynamically adaptive wavelet method for the shallow water equations on a staggered hexagonal C-grid. Pressure is located at the centres of the primal grid (hexagons) and velocity is located at the edges of the dual grid (triangles). Distinct biorthogonal second generation wavelet transforms are developed for the pressure and the velocity. These wavelet transforms are based on second-order accurate interpolation and restriction operators. Together with compatible restriction operators for the mass flux and Bernoulli function, they ensure that mass is conserved and that there is no numerical generation of vorticity when solving the shallow water equations. Grid refinement relies on appropriate thresholding of the wavelet coefficients, allowing error control in both the quasi-geostrophic and inertia-gravity wave regimes. The shallow water equations are discretized on the dynamically adapted multiscale grid using a mass and potential-enstrophy conserving finite-difference scheme. The conservation and error control properties of the method are verified by applying it to a propagating inertia-gravity wave packet and to rotating shallow water turbulence. Significant savings in the number of degrees of freedom are achieved even in the case of rotating shallow-water turbulence. The numerical dissipation introduced by the grid adaptation is quantified. The method has been designed so it can be extended easily to the icosahedral subdivision of the sphere. This work provides important building blocks for the development of fully adaptive general circulation models.

A general framework for high accuracy solutions to energy gradient flows from material science models
Wetton, Brian R.
University of British Columbia

A computational framework is presented for materials science models that come from energy gradient flows that lead to the evolution of structure involving two or more phases. The models are considered in periodic cells and standard Fourier spectral discretization in space is used. Implicit time stepping is used, and the resulting implicit systems are solved iteratively with the preconditioned conjugate gradient method. The dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model (that represents the scaled width of transition layers between phases) is investigated. The framework is easily extended to higher order derivative models, higher dimensional settings, and vector problems. Several examples of its application are demonstrated, including a sixth order problem in three dimensions. A comparison to time-stepping with operator splitting (into convex and concave parts) is done.