Space-time Discretization Research Articles

Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.

Robust approximate symbolic models for a class of continuous-time uncertain nonlinear systems via a control interface

Discrete abstractions have become a standard approach to assist control synthesis under complex specifications. Most techniques for the construction of a discrete abstraction for a continuous-time system require time–space discretization of the concrete system, which constitutes property satisfaction for the continuous-time system non-trivial. In this work, we aim at relaxing this requirement by introducing a control interface. Firstly, we connect the continuous-time uncertain concrete system with its discrete deterministic state-space abstraction with a control interface. Then, a novel stability notion called η-approximately controlled globally practically stable, and a new simulation relation called robust approximate simulation relation are proposed. It is shown that the uncertain concrete system, under the condition that there exists an admissible control interface such that the augmented system (composed of the concrete system and its abstraction) can be made η-approximately controlled globally practically stable, robustly approximately simulates its discrete abstraction. The effectiveness of the proposed results is illustrated by two simulation examples.

Error Estimates for FE-BE Coupling of Scattering of Waves in the Time Domain

Abstract This article considers a coupled finite and boundary element method for an interface problem for the acoustic wave equation. Well-posedness, a priori and a posteriori error estimates are discussed for a symmetric space-time Galerkin discretization related to the energy. Numerical experiments in three dimensions illustrate the performance of the method in model problems.

Reflectionless Klein tunneling of Dirac fermions: comparison of split-operator and staggered-lattice discretization of the Dirac equation

Massless Dirac fermions in an electric field propagate along the field lines without backscattering, due to the combination of spin-momentum locking and spin conservation. This phenomenon, known as ‘Klein tunneling’, may be lost if the Dirac equation is discretized in space and time, because of scattering between multiple Dirac cones in the Brillouin zone. To avoid this, a staggered space-time lattice discretization has been developed in the literature, with one single Dirac cone in the Brillouin zone of the original square lattice. Here we show that the staggering doubles the size of the Brillouin zone, which actually contains two Dirac cones. We find that this fermion doubling causes a spurious breakdown of Klein tunneling, which can be avoided by an alternative single-cone discretization scheme based on a split-operator approach.

Clifford algebra from quantum automata and unitary Wilson fermions

We introduce a spacetime discretization of the Dirac equation that has the form of a quantum automaton and that is invariant upon changing the representation of the Clifford algebra, like the Dirac equation itself. Our derivation follows Dirac's original one: We required that the square of the discrete Dirac scheme be what we define as an acceptable discretization of the Klein-Gordon equation. In contrast to standard lattice gauge theory in discrete time, in which unitarity needs to be proven, we show that the quantum automaton delivers naturally unitary Wilson fermions for any choice of Wilson's parameter.

Sound and soliton wave propagation in homogeneous and heterogeneous mediums with the new two-derivative implicit–explicit Runge–Kutta–Nyström method

This paper derives a new family of implicit–explicit time-marching methods for PDEs with the second-order derivative in time. The present implicit method is based on the two-derivative Runge–Kutta–Nyström methods, which use a third-order time derivative of the solution. Although the current approach is implicit, it does not need to invert the coefficient matrix of the discretized system of equations. The stability properties are assessed using Fourier analysis for the model test problems by considering space–time discretizations together. The present methods are validated by comparing to some of the most widely used time-marching methods available in the literature. In addition, to assess the robustness and efficiency of the present methods, we have also performed numerical simulations of acoustic wave propagation in two- and three-layered heterogeneous media and sine-Gordon solitons for damped and undamped cases. Computed results match very well with the exact and numerical solutions noted in the literature.

Group quantization of the black hole minisuperspace

The emergence of nontrivial symmetries for black holes minisuperspaces has been recently pointed out. These Noether symmetries possess non-null charges and hence map physical solutions to different ones. The symmetry group is isomorphic to the finite-dimensional Poincar\'e group $\mathrm{ISO}(2,1)$, whose irreducible representations are well known. This structure is used to build a consistent quantum theory of black hole minisuperspace. This has, among other consequences, the striking consequence of implying a continuous spectrum for the mass operator. Following loop quantum cosmology, we obtain a regularization scheme compatible with the symmetry structure. It is possible to study the evolution of coherent states following the classical trajectories in the low curvature regime. We show that this produces an effective metric where the singularity is replaced by a Killing horizon merging two asymptotically flat regions. The quantum correction comes from a fundamental discreteness of spacetime, and the uncertainty on the energy of the system. Remarkably, the effective evolution of semiclassical states is described by an effective Hamiltonian, related to the original one through a canonical transformation.

Numerical and convergence analysis of the stochastic Lagrangian averaged Navier–Stokes equations

The primary emphasis of this work is the development of a finite element based space–time discretization for solving the stochastic Lagrangian averaged Navier–Stokes (LANS-α) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of Rd, d∈{2,3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale α, namely we first assume α to be controlled by the step size of the space discretization so that it vanishes when passing to the limit, then we provide an alternative study when α is fixed. A preparatory analysis of uniform estimates in both α and discretization parameters is carried out. Starting out from the stochastic LANS-α model, we achieve convergence toward the continuous strong solutions of the stochastic Navier–Stokes equations in 2D when α vanishes at the limit. Additionally, convergence toward the continuous strong solutions of the stochastic LANS-α model is accomplished if α is fixed. Neither of the mentioned convergences involves the Skorokhod theorem.

Gravitational effects on the Heisenberg Uncertainty Principle: A geometric approach

The Heisenberg Uncertainty Principle (HUP) limits the accuracy in the simultaneous measurements of the position and momentum variables of any quantum system. This is known to be true in the context of non-relativistic quantum mechanics. Based on a semiclassical geometric approach, here we conjecture an effective generalization of this principle, which is well-suited to be extended to general relativity scenarios as well. We apply our formalism to Schwarzschild and de Sitter spacetime, showing that the ensuing uncertainty relations can be mapped into well-known deformations of the HUP. We also infer the form of the perturbed metric that mimics the emergence of a discrete spacetime structure at Planck scale, consistently with the predictions of the Generalized Uncertainty Principle. Finally, we discuss our results in connection with other approaches recently appeared in the literature.

Quantum Gravity If Non-Locality Is Fundamental.

I take non-locality to be the Michelson–Morley experiment of the early 21st century, assume its universal validity, and try to derive its consequences. Spacetime, with its locality, cannot be fundamental, but must somehow be emergent from entangled coherent quantum variables and their behaviors. There are, then, two immediate consequences: (i). if we start with non-locality, we need not explain non-locality. We must instead explain an emergence of locality and spacetime. (ii). There can be no emergence of spacetime without matter. These propositions flatly contradict General Relativity, which is foundationally local, can be formulated without matter, and in which there is no “emergence” of spacetime. If these be true, then quantum gravity cannot be a minor alteration of General Relativity but must demand its deep reformulation. This will almost inevitably lead to: matter not only curves spacetime, but “creates” spacetime. We will see independent grounds for the assertion that matter both curves and creates spacetime that may invite a new union of quantum gravity and General Relativity. This quantum creation of spacetime consists of: (i) fully non-local entangled coherent quantum variables. (ii) The onset of locality via decoherence. (iii) A metric in Hilbert space among entangled quantum variables by the sub-additive von Neumann entropy between pairs of variables. (iv) Mapping from metric distances in Hilbert space to metric distances in classical spacetime by episodic actualization events. (v) Discrete spacetime is the relations among these discrete actualization events. (vi) “Now” is the shared moment of actualization of one among the entangled variables when the amplitudes of the remaining entangled variables change instantaneously. (vii) The discrete, successive, episodic, irreversible actualization events constitute a quantum arrow of time. (viii) The arrow of time history of these events is recorded in the very structure of the spacetime constructed. (ix) Actual Time is a succession of two or more actual events. The theory inevitably yields a UV cutoff of a new type. The cutoff is a phase transition between continuous spacetime before the transition and discontinuous spacetime beyond the phase transition. This quantum creation of spacetime modifies General Relativity and may account for Dark Energy, Dark Matter, and the possible elimination of the singularities of General Relativity. Relations to Causal Set Theory, faithful Lorentzian manifolds, and past and future light cones joined at “Actual Now” are discussed. Possible observational and experimental tests based on: (i). the existence of Sub- Planckian photons, (ii). knee and ankle discontinuities in the high-energy gamma ray spectrum, and (iii). possible experiments to detect a creation of spacetime in the Casimir system are discussed. A quantum actualization enhancement of repulsive Casimir effect would be anti-gravitational and of possible practical use. The ideas and concepts discussed here are not yet a theory, but at most the start of a framework that may be useful.

Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws

We introduce a general framework for enforcing local or global maximum principles in high-order space-time discretizations of a scalar hyperbolic conservation law. We begin with sufficient conditions for a space discretization to be bound preserving (BP) and satisfy a semi-discrete maximum principle. Next, we propose a global monolithic convex (GMC) flux limiter which has the structure of a flux-corrected transport (FCT) algorithm but is applicable to spatial semi-discretizations and ensures the BP property of the fully discrete scheme for strong stability preserving (SSP) Runge–Kutta time discretizations. To circumvent the order barrier for SSP time integrators, we constrain the intermediate stages and/or the final stage of a general high-order RK method using GMC-type limiters. In this work, our theoretical and numerical studies are restricted to explicit schemes which are provably BP for sufficiently small time steps. The new GMC limiting framework offers the possibility of relaxing the bounds of inequality constraints to achieve higher accuracy at the cost of more stringent time step restrictions. The ability of the presented limiters to recognize undershoots/overshoots, as well as smooth solutions, is verified numerically for three representative RK methods combined with weighted essentially nonoscillatory (WENO) finite volume space discretizations of linear and nonlinear test problems in 1D. In this context, we enforce global bounds and prove preservation of accuracy for the linear advection equation.

On two reversible cellular automata with two particle species

We introduce a pair of time-reversible models defined on the discrete space–time lattice with three states per site, specifically, a vacancy and a particle of two flavours (species). The local update rules reproduce the rule 54 reversible cellular automaton when only a single species of particles is present, and satisfy the requirements of flavour exchange (C), space-reversal (P), and time-reversal (T) symmetries. We find closed-form expressions for three local conserved charges and provide an explicit matrix product form of the grand canonical Gibbs states, which are identical for both models. For one of the models this family of Gibbs states seems to be a complete characterisation of equilibrium (i.e. space and time translation invariant) states, while for the other model we empirically find a sequence of local conserved charges, one for each support size larger than 2, hinting to its algebraic integrability. Finally, we numerically investigate the behaviour of spatio-temporal correlation functions of charge densities, and test the hydrodynamic prediction for the model with exactly three local charges. Surprisingly, the numerically observed ‘sound velocity’ does not match the hydrodynamic value. The deviations are either significant, or they decay extremely slowly with the simulation time, which leaves us with an open question for the mechanism of such a glassy behaviour in a deterministic locally interacting system.

Stable Spectral Difference Approach Using Raviart-Thomas Elements for 3D Computations on Tetrahedral Grids

In this paper, the Spectral Difference approach using Raviart-Thomas elements (SDRT) is formulated for the first time on tetrahedral grids. To determine stable formulations, a Fourier analysis is conducted for different SDRT implementations, i.e. different interior flux points locations. This stability analysis demonstrates that using interior flux points located at the Shunn-Ham quadrature rule points leads to linearly stable SDRT schemes up to the third order. For higher orders of accuracy, a significant impact of the position of flux points located on faces is shown. The Fourier analysis is then extended to the coupled time-space discretization and stability limits are determined. Additionally, a comparison between the number of interior FP required for the SDRT scheme and the Flux Reconstruction method is proposed and shows that the two approaches always differ on the tetrahedron. Unsteady validation test cases include a convergence study using the Euler equations and the simulation of the Taylor-Green vortex.

Optimal order multigrid preconditioners for the distributed control of parabolic equations with coarsening in space and time

We devise multigrid preconditioners for linear-quadratic space-time distributed parabolic optimal control problems. While our method is rooted in earlier work on elliptic control, the temporal dimension presents new challenges in terms of algorithm design and quality. Our primary focus is on the cG(s)dG(r) discretizations which are based on functions that are continuous in space and discontinuous in time, but our technique is applicable to various other space-time finite element discretizations. We construct and analyse two kinds of multigrid preconditioners: the first is based on full coarsening in space and time, while the second is based on semi-coarsening in space only. Our analysis, in conjunction with numerical experiments, shows that both preconditioners are of optimal order with respect to the discretization in case of cG(1)dG(r) for r = 0, 1 and exhibit a suboptimal behaviour in time for Crank–Nicolson. We also show that, under certain conditions, the preconditioner using full space-time coarsening is more efficient than the one involving semi-coarsening in space, a phenomenon that has not been observed previously. Our numerical results confirm the theoretical findings.

Highly accurate quadrature schemes for singular integrals in energetic BEM applied to elastodynamics

We consider an exterior linear elastodynamics problem with vanishing initial conditions and Dirichlet datum on the scatterer. We convert the Navier Equation, governing the wave behaviour, into two space–time Boundary Integral Equations (BIEs) whose solution is approximated by the energetic Boundary Element Method (BEM). To apply this technique, we have to set the BIEs in a weak form related to the energy of the differential problem solution at the final time instant of analysis. After the space–time discretization of the weak formulation, we have to deal with double space–time integrals, with a weakly singular kernel depending on primary and secondary wave speeds and multiplied by Heaviside functions. The main purpose of this work is the analysis of these peculiar integrals and the study of suitable quadrature schemes for their approximation.

A discrete relativistic spacetime formalism for 1 + 1-QED with continuum limits

We build a quantum cellular automaton (QCA) which coincides with 1+1 QED on its known continuum limits. It consists in a circuit of unitary gates driving the evolution of particles on a one dimensional lattice, and having them interact with the gauge field on the links. The particles are massive fermions, and the evolution is exactly U(1) gauge-invariant. We show that, in the continuous-time discrete-space limit, the QCA converges to the Kogut–Susskind staggered version of 1+1 QED. We also show that, in the continuous spacetime limit and in the free one particle sector, it converges to the Dirac equation—a strong indication that the model remains accurate in the relativistic regime.

Strong rates of convergence of space-time discretization schemes for the 2D Navier–Stokes equations with additive noise

We consider the strong solution of the 2D Navier–Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the space-time rate of convergence is the “optimal” one, namely, [Formula: see text] in time and 1 in space. Let us mention that the coefficient [Formula: see text] is equal to the time regularity of the solution with values in [Formula: see text]. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Unlike previous results, our main new idea is the use of a discrete Grönwall lemma for the error estimate without any localization.

Finite-difference wave-propagation models for dispersive media: impact of space-time discretization

PurposeThe computational accuracy and performance of finite-difference time-domain (FDTD) methods are affected by the implementation of approximating derivative formulae in diverse ways. This study aims to focus on FDTD models featuring material dispersion with negligible losses and investigates two specific aspects that, until today, are usually examined in the context of non-dispersive media only. These aspects pertain to certain abnormal characteristics of coarsely resolved electromagnetic waves and the selection of the proper time-step size, in the case of a high-order discretization scheme.Design/methodology/approachConsidering a Lorentz medium with negligible losses, the propagation characteristics of coarsely resolved waves is examined first, by investigating thoroughly the numerical dispersion relation of a typical discretization scheme. The second part of the study is related to the unbalanced space-time errors in FDTD schemes with dissimilar space-time approximation orders. The authors propose a remedy via the suitable choice of the time-step size, based on the single-frequency minimization of an error expression extracted, again, from the scheme’s numerical dispersion formula.FindingsUnlike wave propagation in free space, there exist two parts of the frequency spectrum where waves in a Lorentz medium experience non-physical attenuation and display non-changing propagation constants, due to coarse discretization. The authors also show that an optimum time-step size can be determined, in the case of the (2,4) FDTD scheme, which minimizes the selected error formula at a specific frequency point, promoting more efficient implementations.Originality/valueUnique characteristics displayed by discretized waves, which have been known for non-dispersive media, are examined and verified for the first time in the case of dispersive materials, thus completing the comprehension of the space-time discretization impact on simulated quantities. In addition, the closed-form formula of the optimum time-step enables the efficient implementation of the (2,4) FDTD method, minimizing the detrimental influence of the low-order temporal integration.

A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

A new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step Delta t satisfy hcong CDelta t, with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the L^infty and L^2-norms is order p+1 for polynomial orders p=1 and p=3 and order p for polynomial order p=2.

Kernel representation of Kalman observer and associated H-matrix based discretization

In deterministic estimation, applying a Kalman filter to a dynamical model based on partial differential equations is theoretically seducing but solving the associated Riccati equation leads to a so-called curse of dimensionality for its numerical implementation. In this work, we propose to entirely revisit the theory of Kalman filters for parabolic problems where additional regularity results proves that the Riccati equation solution belongs to the class of Hilbert-Schmidt operators. The regularity of the associated kernel then allows to proceed to the numerical analysis of the Kalman full space-time discretization in adapted norms, hence justifying the implementation of the related Kalman filter numerical algorithm with H-matrices typically developed for integral equations discretization.