Higher-order Extension Research Articles

Efficient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions

The discontinuous Galerkin (DG) methods provide a high-order extension of the finite volume method in much the same way as high-order or spectral/hp elements extend standard finite elements. However, lack of inter-element continuity is often contrary to the smoothness assumptions upon which many post-processing algorithms such as those used in visualization are based. Smoothness-increasing accuracy-conserving (SIAC) filters were proposed as a means of ameliorating the challenges introduced by the lack of regularity at element interfaces by eliminating the discontinuity between elements in a way that is consistent with the DG methodology; in particular, high-order accuracy is preserved and in many cases increased. The goal of this paper is to explicitly define the steps to efficient computation of this filtering technique as applied to both structured triangular and quadrilateral meshes. Furthermore, as the SIAC filter is a good candidate for parallelization, we provide, for the first time, results that confirm anticipated performance scaling when parallelized on a shared-memory multi-processor machine.

A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids

We present a one-step high-order cell-centered numerical scheme for solving Lagrangian hydrodynamics equations on unstructured grids. The underlying finite volume discretization is constructed through the use of the sub-cell force concept invoking conservation and thermodynamic consistency. The high-order extension is performed using a one-step discretization, wherein the fluxes are computed by means of a Taylor expansion. The time derivatives of the fluxes are obtained through the use of a node-centered solver which can be viewed as a two-dimensional extension of the Generalized Riemann Problem methodology introduced by Ben-Artzi and Falcovitz.

Characterization of higher-order monotonicity via integral inequalities

The Hermite-Hadamard inequality not only is a consequence of convexity but also characterizes it: if a continuous function satisfies either its left-hand side or its right-hand side on each compact subinterval of the domain, then it is necessarily convex. The aim of this paper is to prove analogous statements for the higher-order extensions of the Hermite-Hadamard inequality. The main tools of the proofs are smoothing by convolution and the support properties of higher-order monotone functions.

A local greedy algorithm and higher-order extensions for global numerical continuation of analytically varying subspaces

We present a family of numerical implementations of Kato’s ODE propagating global bases of analytically varying invariant subspaces of which the first-order version is a surprisingly simple “greedy algorithm” that is both stable and easy to program and the second-order version a relaxation of a first-order scheme of Brin and Zumbrun. The method has application to numerical Evans function computations used to assess stability of traveling-wave solutions of time-evolutionary PDE.

A General Index of Absolute Risk Attitude

Many results involving expected utility theory call upon the notions of absolute risk aversion, prudence, and/or temperance. This paper exploits a representation of the Friedman–Savage utility premium (Friedman, M., L. J. Savage. 1948. The utility analysis of choices involving risk. J. Political Econom. 56(4) 279–304) to give a general foundation for such coefficients and for their higher-order extensions.

On the accuracy of the finite element method plus time relaxation

If u denotes a local, spatial average of u, then u' = u — u is the associated fluctuation. Consider a time relaxation term added to the usual finite element method. The simplest case for the model advection equation u t +a · ∇u = (x, t) is (u h,t + a · ∇u h ,v h )+χ(u' h ,v' h )=(f(x,t),v h ). We analyze the error in this and (more importantly) higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation.

Compressible Boundary-Layer Predictions at High Reynolds Number Using Hybrid LES/RANS Methods

Simulations of compressible boundary-layer flow at three different Reynolds numbers (Re δ = 5.59 × 10 4 , 1.78 × 10 5 , and 1.58 x 10 6 ) are performed using a hybrid large-eddy simulation/Reynolds-averaged Navier-Stokes method. Variations in the recycling/rescaling method, the higher order extension, the choice of primitive variables, the Reynolds-averaged Navier-Stokes to large eddy simulation transition parameters, and the mesh resolution are considered in order to assess the model. The results indicate that the present model can provide good predictions of the mean-flow properties, second-moment statistics, and structural features of the boundary layers considered. Normalized turbulent statistics in the outer layer are found to be independent of Reynolds number, similar to incompressible turbulent boundary layers.

Tensor Decompositions and Applications

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N-way array. Decompositions of higher-order tensors (i.e., N-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric conservation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area-weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the other hand, the area-weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equi-angular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.

Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme

A finite volume well-balanced weighted essentially nonoscillatory (WENO) scheme, fourth-order accurate in space and time, for the numerical integration of shallow water equations with the bottom slope source term, is presented. The main novelty introduced in this work is a new method for managing bed discontinuities. This method is based on a suitable reconstruction of the conservative variables at the cell interfaces, coupled with a correction of the numerical flux based on the local conservation of total energy. Further changes regard the treatment of the source term, based on a high-order extension of the divergence form for bed slope source term method, and the application of an analytical inversion of the specific energy-depth relationship. Two ad hoc test cases, consisting of a steady flow over a step and a surge crossing a step, show the effectiveness of the method of treating bottom discontinuities. Several standard one-dimensional test cases are also used to verify the high-order accuracy, the C-...

Velocity-Scheduling Control for a Unicycle Mobile Robot: Theory and Experiments

Improvement over classical dynamic feedback linearization for a unicycle mobile robots is proposed. Compared to classical extension, the technique uses a higher-dimensional state extension, which allows rejecting a constant disturbance on the robot rotational axis. The proposed dynamic extension acts as a velocity scheduler that specifies, at each time instant, the ideal translational velocity that the robot should have. By using a higher-order extension, both the magnitude and the orientation of the velocity vector can be generated, which introduces robustness in the control scheme. Stability for both asymptotic convergence to a point and trajectory tracking is proven. The theoretical results are illustrated first in simulation, and then experimentally on the autonomous mobile robot <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fouzy</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">III</i> .

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.

Logical Extensions of Aristotle’s Square

We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle’s square under its modal form has the following four vertices: A is □α, E is $$\square\neg\alpha$$ , I is $$\neg\square\neg\alpha$$ and O is $$\square\neg\alpha$$ , where α is a logical formula and □ is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether $$\neg$$ is involutive or not) modal logic. [3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of $$\{\alpha,\square\alpha\}$$ by the logical operations $$\{\neg,\wedge,\vee\}$$ , under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle’s square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures.

Generalized Symmetries of Massless Free Fields on Minkowski Space

A complete and explicit classification of generalized, or local, symmetries of massless free fields of spin s 1/2 is carried out. Up to equivalence, these are found to consists of the conformal symmetries and their duals, new chiral symmetries of order 2s, and their higher-order extensions obtained by Lie differentiation with respect to confor- mal Killing vectors. In particular, the results yield a complete classification of generalized symmetries of the Dirac-Weyl neutrino equation, Maxwell's equations, and the linearized gravity equations.

Finite Volume Simulation of the Geostrophic Adjustment in a Rotating Shallow-Water System

The goal of this article is to simulate rotating flows of shallow layers of fluid by means of finite volume numerical schemes. More precisely, we focus on the simulation of the geostrophic adjustment phenomenon. As spatial discretization, a first order Roe-type method and some higher-order extensions are developed. The time discretization is designed in order to provide suitable approximations of inertial oscillations, taking into account the Hamiltonian structure of the system for these solutions. The numerical dispersion laws and the wave amplifications of the schemes are studied, and their well-balanced properties are analyzed. Finally, some numerical experiments for one-dimensional (1d) and two-dimensional (2d) problems are shown.

Well-Balanced High Order Extensions of Godunov's Method for Semilinear Balance Laws

This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunov's methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunov's methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes.

Increasing the accuracy in locally divergence-preserving finite volume schemes for MHD

It is of utmost interest to control the divergence of the magnetic flux in simulations of the ideal magnetohydrodynamic equations since, in general, divergence errors tend to accumulate and render the schemes unstable. This paper presents a higher-order extension of the locally divergence-preserving procedure developed in Torrilhon [M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comput. 26 (2005) 1166–1191]; a fourth-order accurate local redistribution of the numerical magnetic field fluxes of a finite volume base scheme is introduced. The redistribution ensures that a fourth-order accurate discrete divergence operator is preserved to round off errors when applied to the cell averages of the magnetic flux density. The developed procedure is applicable to generic semi-discrete finite volume schemes and its purpose is to stabilize the schemes using a local procedure that respects the accuracy of the base scheme to a greater extent than the previous second-order achievements. Numerical experiments that demonstrate the properties of the new procedure are also presented.

A discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid/solid problems

An evanescent wave is produced when a propagating incident wave impinges on an interface between two media or materials at a sub-critical angle. The sub-scale nature of such a wave makes it difficult to be captured computationally. In this paper, the Discontinuous Enrichment Method (DEM) developed in [C. Farhat, I. Harari, L.P. Franca, The discontinuous enrichment method, Comput. Methods Appl. Mech. Engrg. 190 (2001) 6455–6479; C. Farhat, I. Harari, U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appl. Mech. Engrg. 192 (2003) 1389–1419; C. Farhat, I. Harari, U. Hetmaniuk, The discontinuous enrichment method for multiscale analysis, Comput. Methods Appl. Mech. Engrg. 192 (2003) 3195–3210; C. Farhat, R. Tezaur, P. Weidemann-Goiran, Higher-order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems, Int. J. Numer. Methods Engrg. 61 (2004) 1938–1956; C. Farhat, P. Wiedemann-Goiran, R. Tezaur, A discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of short wave exterior Helmholtz problems on unstructured meshes. New computational methods for wave propagation, Wave Motion 39(4) (2004) 307–317; L. Zhang, R. Tezaur, C. Farhat, The discontinuous enrichment method for elastic wave propagation in the medium-frequency regime, Int. J. Numer. Methods Engrg. 66 (2006) 2086–2114; R. Tezaur, C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, Int. J. Numer. Methods Engrg. 66 (2006) 796–815] is extended to a class of evanescent wave problems. New DEM elements are constructed by enriching polynomial approximations with free-space evanescent solutions in order to achieve high accuracy for problems with fluid/fluid or fluid/solid interfaces on which evanescent waves may occur. The new DEM elements are applied to the solution of various two-dimensional model problems with evanescent waves. Their performance is observed to be better than that of the basic Helmholtz type DEM elements, and superior to that of the classical higher-order polynomial finite element method.

On the formulations of higher-order strain gradient crystal plasticity models

Recently, several higher-order extensions to the crystal plasticity theory have been proposed to incorporate effects of material length scales that were missing links in the conventional continuum mechanics. The extended theories are classified into work-conjugate and non-work-conjugate types. A common feature of the former is that existence of higher-order stresses work-conjugate to gradients of plastic strain is presumed and an extended principle of virtual work involving such an additional virtual work contribution is formulated. Meanwhile, in the latter type, the higher-order stress quantities do not appear explicitly. Instead, rates of crystallographic slip are influenced by back stresses that arise in response to spatial gradients of the geometrically necessary dislocation densities. The work-conjugate type and the non-work-conjugate type of theories have different theoretical backgrounds and very unlike mathematical representations. Nevertheless, both types of theories predict the same kind of material length scale effects. We have recently shown that there exists some equivalency between the two approaches in the special situation of two-dimensional single slip under small deformation. In this paper, the discussion is extended to a more general situation, i.e. the context of multiple and three-dimensional slip deformations.

Self-Similarity: Part II—Optimal Estimation of Fractal Processes

In a companion paper (see Self-Similarity: Part I-Splines and Operators), we characterized the class of scale-invariant convolution operators: the generalized fractional derivatives of order gamma. We used these operators to specify regularization functionals for a series of Tikhonov-like least-squares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of these smoothing splines and proposed a fast Fourier transform (FFT)-based implementation. Here, we present an alternative stochastic formulation to further justify these fractional spline estimators. As suggested by the title, the relevant processes are those that are statistically self-similar; that is, fractional Brownian motion (fBm) and its higher order extensions. To overcome the technical difficulties due to the nonstationary character of fBm, we adopt a distributional formulation due to Gel'fand. This allows us to rigorously specify an innovation model for these fractal processes, which rests on the property that they can be whitened by suitable fractional differentiation. Using the characteristic form of the fBm, we then derive the conditional probability density function (PDF) p(BH(t)|Y), where Y={BH(k)+n[k]}kisinZ are the noisy samples of the fBm BH(t) with Hurst exponent H. We find that the conditional mean is a fractional spline of degree 2H, which proves that this class of functions is indeed optimal for the estimation of fractal-like processes. The result also yields the optimal [minimum mean-square error (MMSE)] parameters for the smoothing spline estimator, as well as the connection with kriging and Wiener filtering