Conservative space-time mesh refinement methods for the FDTD solution of Maxwell’s equations

A new variational space-time mesh refinement method is proposed for the FDTD solution of Maxwell’s equations. The main advantage of this method is to guarantee the conservation of a discrete energy that implies that the scheme remains L² stable under the usual CFL condition. The only additional cost induced by the mesh refinement is the inversion, at each time step, of a sparse symmetric positive definite linear system restricted to the unknowns located on the interface between coarse and fine grid. The method is presented in a rather general way and its stability is analyzed. An implementation is proposed for the Yee scheme. In this case, various numerical results in 3-D are presented in order to validate the approach and illustrate the practical interest of space-time mesh refinement methods.     

Schwarz method for earthquake source dynamics

Dynamic faulting under slip-dependent friction in a linear elastic domain (in-plane and 3D configurations) is considered. The use of an implicit time-stepping scheme (Newmark method) allows much larger values of the time step than the critical CFL time step, and higher accuracy to handle the non-smoothness of the interface constitutive law (slip weakening friction).The finite element form of the quasi-variational inequality is solved by a Schwarz domain decomposition method, by separating the inner nodes of the domain from the nodes on the fault. In this way, the quasi-variational inequality splits into two subproblems. The first one is a large linear system of equations, and its unknowns are related to the mesh nodes of the first subdomain (i.e. lying inside the domain). The unknowns of the second subproblem are the degrees of freedom of the mesh nodes of the second subdomain (i.e. lying on the domain boundary where the conditions of contact and friction are imposed). This nonlinear subproblem is solved by the same Schwarz algorithm, leading to some local nonlinear subproblems of a very small size.Numerical experiments are performed to illustrate convergence in time and space, instability capturing, energy dissipation and the influence of normal stress variations. We have used the proposed numerical method to compute source dynamics phenomena on complex and realistic 2D fault models (branched fault systems).     

High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics

We present a class of hybridizable discontinuous Galerkin (HDG) methods for the numerical simulation of wave phenomena in acoustics and elastodynamics. The methods are fully implicit and high-order accurate in both space and time, yet computationally attractive owing to their following distinctive features. First, they reduce the globally coupled unknowns to the approximate trace of the velocity, which is defined on the element faces and single-valued, thereby leading to a significant saving in the computational cost. In addition, all the approximate variables (including the approximate velocity and gradient) converge with the optimal order of k + 1 in the L²-norm, when polynomials of degree k ? 0 are used to represent the numerical solution and when the time-stepping method is accurate with order k + 1. When the time-stepping method is of order k + 2, superconvergence properties allows us, by means of local postprocessing, to obtain better, yet inexpensive approximations of the displacement and velocity at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k + 2 in the L²-norm when k ? 1 for both acoustics and elastodynamics. Extensive numerical results are provided to illustrate these distinctive features.     

An analysis of time discretization in the finite element solution of hyperbolic problems

The problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined. A modified equation analysis reveals that the classical, second-order accurate, time-stepping algorithms, i.e., the Lax-Wendroff, leap-frog, and Crank-Nicolson methods, properly combine with piecewise linear finite elements in advection problems only for small values of the time step. On the contrary, as the Courant number increases, the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit CFL property are not achieved. These fundamental numerical properties can, however, be recovered, while still remaining in the standard Galerkin finite element setting, by increasing the order of accuracy of the time discretization. This is accomplished by exploiting the Taylor series expansion in the time increment up to the third order before performing the Galerkin spatial discretization using piecewise linear interpolations. As a result, it appears that the proper finite element equivalents of second-order finite difference schemes are implicit methods of incremental type having third- and fourth-order global accuracy on uniform meshes (Taylor-Galerkin methods). Numerical results for several linear examples are presented to illustrate the properties of the Taylor-Galerkin schemes in one- and two-dimensional calculations.     

A numerical study of Langevin QCD with dynamical quarks

QCD with four flavours of dynamical staggered quarks of mass ma=0.1 is simulated on a 4⁴ lattice, using two versions of the second order Langevin algorithm with bilinear noise: the naive version and one which compensates for the non-integrable term which appears to first order in the discrete Langevin time step size. Comparison with the results of an exact numerical computation of the fermion determinant reveals that these algorithms yield accurate results only at rather small values of the step size. The correction due to the non-integrable term is quantitatively unimportant for such step sizes.     

On the relation between FDTD and Fibonacci polynomials

In this paper we show that the Finite-Difference Time-Domain method (FDTD method) follows the recurrence relation for Fibonacci polynomials. More precisely, we show that FDTD approximates the electromagnetic field by Fibonacci polynomials in ΔtA, where Δt is the time step and A is the first-order Maxwell system matrix. By exploiting the connection between Fibonacci polynomials and Chebyshev polynomials of the second kind, we easily obtain the Courant-Friedrichs-Lewy (CFL) stability condition and we show that to match the spectral width of the system matrix, the time step should be chosen as large as possible, that is, as close to the CFL upper bound as possible.     

Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

The dynamics of an electron gas in a constant ion background can be decribed by the Vlasov-Poisson-Boltzmann system at the kinetic level, or by the compressible Euler-Poisson system at the fluid level. We prove that any solution of the Vlasov-Poisson-Boltzmann system near a smooth local Maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the Euler-Poisson system, as the mean free path ε goes to zero. We use a recent L ² − L ^∞ framework in the Boltzmann theory to control the higher order remainder in the Hilbert expansion uniformly in ε and globally in time.     

Minimality via Second Variation for a Nonlocal Isoperimetric Problem

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L ¹-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.     

Global Solutions to Gross–Neveu Equation

We prove global existence of solutions to Gross–Neveu equations. Given a local solution, we obtain a uniform L ^∞ bound of the solution by applying local form of charge conservation.     

Self-similarity and asymptotic stability for coupled nonlinear Schrödinger equations in high dimensions

This paper is concerned with systems of coupled Schrödinger equations with polynomial nonlinearities and dimension n≥1. We show the existence of global self-similar solutions and prove that they are asymptotically stable in a framework based on weak-L^p spaces, whose elements have local finite L²-mass. The radial symmetry of the solutions is also addressed.     

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

In this work we show that the flexibility of the discontinuous Galerkin (dG) discretization can be fruitfully exploited to implement numerical solution strategies based on the use of elements with very general shapes. Thanks to the freedom in defining the mesh topology, we propose a new h-adaptive technique based on agglomeration coarsening of a fine mesh. The possibility to enhance the error distribution over the computational domain is investigated on a Poisson problem with the goal of obtaining a mesh independent discretization.The main building block of our dG method consists of defining discrete polynomial spaces directly on physical frame elements. For this purpose we orthonormalize with respect to the L²-product a set of monomials relocated in a specific element frame and we introduce an easy way to reduce the cost related to numerical integration on agglomerated meshes. To complete the dG formulation for second order problems, two extensions of the BR2 scheme to arbitrary polyhedral grids, including an estimate of the stabilization parameter ensuring the coercivity property, are here proposed.     

Interconnect and lumped elements modeling in interior penalty discontinuous Galerkin time-domain methods

In this paper, we present an approach on how to incorporate passive lumped elements such as resistors, capacitors and inductors in DGTD methods and their application to interconnect modeling. Starting from the voltage and current relationships, we derive the equivalent relationships that describe each of the R, L, C in terms of the electric and magnetic fields. Next, these field expressions are weakly enforced through the Interior Penalty (IP) DG formulation. The proposed method is explicit and conditionally stable. Additionally, a local time-stepping strategy is applied to increase efficiency and reduce the computational time. Finally, a numerical example is presented to validate the proposed approach.     

Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers

We present a robust and accurate numerical algorithm for calculating energy-minimizing wavelengths of equilibrium states for diblock copolymers. The phase-field model for diblock copolymers is based on the nonlocal Cahn–Hilliard equation. The model consists of local and nonlocal terms associated with short- and long-range interactions, respectively. To solve the phase-field model efficiently and accurately, we use a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method. To find energy-minimizing wavelengths of equilibrium states, we take two approaches. One is to obtain an equilibrium state from a long time simulation of the time-dependent partial differential equation with varying periodicity and choosing the energy-minimizing wavelength. The other is to directly solve the ordinary differential equation for the steady state. The results from these two methods are identical, which confirms the accuracy of the proposed algorithm. We also propose a simple and powerful formula: h = L¹/m, where h is the space grid size, L¹ is the energy-minimizing wavelength, and m is the number of the numerical grid steps in one period of a wave. Two- and three-dimensional numerical results are presented validating the usefulness of the formula without trial and error or ad hoc processes.     

Existence,Non-existence,and Uniqueness for a Heat Equation with Exponential Nonlinearity in ℝ2

We consider a semilinear heat equation with exponential nonlinearity in ?². We prove that local solutions do not exist for certain data in the Orlicz space exp L ²(?²), even though a small data global existence result holds in the same space exp L ²(?²). Moreover, some suitable subclass of exp L ²(?²) for local existence and uniqueness is proposed.

     

Failsafe flux limiting and constrained data projections for equations of gas dynamics

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L² projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L² projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples.     

Non-Ergodicity in a 1-D Particle Process with Variable Length

We present a 1-D random particle process with uniform local interaction, which displays some form of non-ergodicity, similar to contact processes, but more unexpected. Particles, enumerated by integer numbers, interact at every step of the discrete time only with their nearest neighbors. Every particle has two possible states, called minus and plus. At every time step two transformations occur. The first one turns every minus into plus with probability β independently from what happens at other places and thereby favors pluses against minuses. The second one is “impartial.” Under its action, whenever a plus is a left neighbor of a minus, both disappear with probability α independently from presence and fate of other pairs of this sort. If β is small enough by comparison with α ² and we start with “all minuses,” the minuses never die out.     

Classical Hamiltonian perturbation theory without secular terms or small denominators

We consider perturbation theory in ? for the classical Hamiltonian H = H₀ + ?H₁, where H₀ gives rise to a known motion and ? is small. First we demonstrate how the usual secular terms and small denominators arise from a straightforward expansion in ? and argue that they are artifacts of the method. Then we present an alternative perturbation theory based on an analysis of the operator (s ? L)^?1, where s is a complex number and L is the Liouville operator corresponding to H. This perturbation series contains neither secular terms nor small denominators. In the case of almost multiply periodic systems we show, to lowest non-trivial order in ?, how our series reproduces the standard results both in the resonant and nonresonant regions — all in one analytic formula. As a final exercise we demonstrate that energy is conserved at order ?ⁿ⁺¹ when the accuracy of the theory is order ?ⁿ.     

An efficient method for the incompressible Navier–Stokes equations on irregular domains with no-slip boundary conditions,high order up to the boundary

Common efficient schemes for the incompressible Navier–Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier–Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, including semi-implicit treatments of viscosity, and in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the L^∞ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D.     

Efficient symmetric positive definite second-order accurate monolithic solver for fluid/solid interactions

We introduce a robust and efficient method to simulate strongly coupled (monolithic) fluid/rigid-body interactions. We take a fractional step approach, where the intermediate state variables of the fluid and of the solid are solved independently, before their interactions are enforced via a projection step. The projection step produces a symmetric positive definite linear system that can be efficiently solved using the preconditioned conjugate gradient method. In particular, we show how one can use the standard preconditioner used in standard fluid simulations to precondition the linear system associated with the projection step of our fluid/solid algorithm. Overall, the computational time to solve the projection step of our fluid/solid algorithm is similar to the time needed to solve the standard fluid-only projection step. The monolithic treatment results in a stable projection step, i.e. the kinetic energy does not increase in the projection step. Numerical results indicate that the method is second-order accurate in the L^∞-norm and demonstrate that its solutions agree quantitatively with experimental results.