Low-order Schemes Research Articles

Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems

The algebraic flux correction (AFC) schemes presented in this work constrain a standard continuous finite element discretization of a nonlinear hyperbolic problem to satisfy relevant maximum principles and entropy stability conditions. The desired properties are enforced by applying a limiter to antidiffusive fluxes that represent the difference between the high-order baseline scheme and a property-preserving approximation of Lax–Friedrichs type. In the first step of the limiting procedure, the given target fluxes are adjusted in a way that guarantees preservation of local and/or global bounds. In the second step, additional limiting is performed, if necessary, to ensure the validity of fully discrete and/or semi-discrete entropy inequalities. The limiter-based entropy fixes considered in this work are applicable to finite element discretizations of scalar hyperbolic equations and systems alike. The underlying inequality constraints are formulated using Tadmor’s entropy stability theory.The proposed limiters impose entropy-conservative or entropy-dissipative bounds on the rate of entropy production by antidiffusive fluxes and Runge–Kutta (RK) time discretizations. Two versions of the fully discrete entropy fix are developed for this purpose. The first one incorporates temporal entropy production into the flux constraints, which makes them more restrictive and dependent on the time step. The second algorithm interprets the final stage of a high-order AFC-RK method as a constrained antidiffusive correction of an implicit low-order scheme (algebraic Lax–Friedrichs in space + backward Euler in time). In this case, iterative flux correction is required, but the inequality constraints are less restrictive and limiting can be performed using algorithms developed for the semi-discrete problem. To motivate the use of limiter-based entropy fixes, we prove a finite element version of the Lax–Wendroff theorem and perform numerical studies for standard test problems. In our numerical experiments, entropy-dissipative schemes converge to correct weak solutions of scalar conservation laws, of the Euler equations, and of the shallow water equations.

Inexact accelerated high-order proximal-point methods

In this paper, we present a new framework of bi-level unconstrained minimization for development of accelerated methods in Convex Programming. These methods use approximations of the high-order proximal points, which are solutions of some auxiliary parametric optimization problems. For computing these points, we can use different methods, and, in particular, the lower-order schemes. This opens a possibility for the latter methods to overpass traditional limits of the Complexity Theory. As an example, we obtain a new second-order method with the convergence rate Oleft( k^{-4}right) , where k is the iteration counter. This rate is better than the maximal possible rate of convergence for this type of methods, as applied to functions with Lipschitz continuous Hessian. We also present new methods with the exact auxiliary search procedure, which have the rate of convergence Oleft( k^{-(3p+1)/ 2}right) , where p ge 1 is the order of the proximal operator. The auxiliary problem at each iteration of these schemes is convex.

Throughput‐based energy and spectral efficiency tradeoff for massive MIMO systems

In massive MIMO systems, energy efficiency (EE) and spectral efficiency (SE) can be balanced by tuning the number of the transmit antennas and the transmit power. However, two key parameters i.e. the modulation order and the bit-error-rate (BER) are seldom considered in the conventional EE-SE tradeoff for the massive MIMO systems. To address this problem, the throughput-based EE (TEE) and SE (TSE) tradeoff for both the conventional massive MIMO and the full-dimensional (FD) massive MIMO systems are investigated. Firstly, a TEE and TSE tradeoff problem is formulated and the transition property of the modulation order is analysed. The analysis reveals that due to the discrete property of the modulation order, the TEE-TSE tradeoff curve is continuously non-differentiable at the transition point of the modulation order. By utilising the transition property, a modified normal-boundary-intersection particle-swarm-optimisation (NBI-PSO) algorithm is developed to solve the TEE-TSE problem effectively. From the numerical results, it is found that the throughput-based problem has a relatively smaller EE and SE on the Pareto front than the capacity-based one and selecting an improper combination of the modulation order, the transmit power and the number of the antennas may cause the EE-SE relation sub-optimal from the point view of Pareto efficiency. Through the numerical results, the most energy efficient modulation order is obtained and it is not the commonly assumed low-order modulation scheme.

Integral barrier Lyapunov function-based three-dimensional low-order integrated guidance and control design with seeker's field-of-view constraint

As the strap-down seeker with its detector is fixed onto the body of the flight vehicle, its measurement is coupled with the information of the attitude motion of the vehicle's body, which results that the inertial line-of-sight (LOS) angular rate used in the traditional guidance laws cannot be measured. Besides, the process that the vehicle adjusts its attitude to realize the instantaneous overload command may cause the target to be outside the field-of-view (FOV) of the seeker. To deal with the FOV constraint and the lack of the inertial LOS angular rate, the novel body-LOS coordinate system based relative motion equations are derived, in which the derivative of attitude angular rates appears. Combined with the dynamics of attitude angular rates, the original low-order integrated guidance and control (IGC) design model with FOV constraint and the roll channel IGC design model are established. Furthermore, an integral Barrier Lyapunov function (IBLF) based three-dimensional low-order IGC scheme with the FOV constraint is proposed in this paper. Meanwhile, the adaptive algorithms are employed to estimate the square of the upper bound of the disturbance, which would be compensated in the IGC scheme to improve the robustness of the control system. Both system order and tuning parameters of the proposed low-order IGC algorithm dramatically decrease, and the design procedure is significantly simplified. With the aid of Lyapunov stability analysis, states of the IGC closed-loop system are proved to be uniformly ultimate bounded. Finally, extensive numerical simulations are carried out to validate the effectiveness and robustness of the newly proposed IGC scheme.

On the numerical capture of Taylor column phenomena in rotating viscous fluid

Over the years, a number of researchers have explored the possibilities of capturing the fantastic Taylor column phenomena numerically. However, comprehensive satisfactory numerical results have not been obtained in spite of the presence of well established experimental and theoretical data. One important blockade in the process is that the existing lower-order schemes cannot capture the phenomena appropriately owing to the complexity of the flow which arises due to the effect of several parameters and nonlinearities. Hence we propose to use the Higher Order Compact Scheme (HOCS) to capture the Taylor column phenomena along with all its essential features. The results obtained using HOCS are found to be in line with available experimental results in the literature. The physical analysis of streamlines, swirl velocity, surface pressure, and pressure gradient show that there arises a reverse flow near the front and a cyclonic vortex near the rear stagnation point along with a geostrophic region under certain conditions depending on the balance between the Coriolis force and the pressure gradient.

A hybrid first-order and WENO scheme for the high-resolution and computationally efficient modeling of pollutant transport

To improve the modeling quality of pollutant transport in shallow waters, different reconstruction schemes have been proposed to better link the edge values to the centroid values of a pollutant concentration in finite-volume shallow water models: a scheme of higher (lower) order generally has a better (poorer) quantitative accuracy but lower (higher) computational efficiency. Here, a numerical comparative study of several classical schemes is first conducted under a variety of pollutant distribution conditions. The results reveal that, for the condition of relatively uniform pollutant distribution, the numerical accuracy of a lower-order scheme (such as the first-order scheme or the MUSCL scheme) may be similar to that of a higher-order scheme (such as the WENO scheme). The second-order derivative of the concentration, here termed the nonlinear indicator (NI), correlates well with the discrepancies between the numerical solutions and analytical solutions. A threshold value of approximately 10−7∼10−6 m-2 for the NI is identified, above which a higher-order scheme may be required. Based on this understanding, a hybrid first-order and WENO scheme is proposed. Numerical case studies show that the hybrid scheme can successfully combine the efficiency of the first-order scheme with the high accuracy of the WENO scheme for pollutant modeling.

Comparison and refinement of the various full-spectrum k-distribution and spectral line weighted-sum-of-gray-gases models for nonhomogeneous media

Both the full-spectrum k-distribution (FSK) method and the spectral line weighted-sum-of-gray-gases (SLW) method are global spectral methods, with the former based on a spectral reordering concept, while the latter uses spectral binning. Both can provide excellent accuracy with outstanding numerical efficiency to model radiative heat transfer in combustion gases. In the present paper we aim to complete the development of both methods, and to tie them together, pointing out their commonalities and differences of their nonhomogeneous media extensions, employing either the correlated or scaled absorption coefficient assumption. Results from the complete set of plausible implementations are discussed and compared in detail for several radiative heat transfer calculations carried out in both 1D slabs and a real combustion field. The results show that emission is an important criterion to examine the accuracy of global methods applied to nonhomogeneous media, i.e., those that preserve emission generally give more accurate results than those that do not. It was also found that, while proper choice of the reference temperature required by all methods is important, the recommended methods appear to be only weakly dependent on that choice. Finally, equivalent SLW schemes are generally somewhat less accurate than their FSK counterparts due to their low-order spectral integration scheme; and this may be exacerbated if full-spectrum k-distributions are determined by mixing of values from individual species, as appears to be the preferred approach by SLW users to date.

The role of the dual grid in low-order compatible numerical schemes on general meshes

In this work, we uncover hidden geometric aspect of low-order compatible numerical schemes. First, we rewrite standard mimetic reconstruction operators defined by Stokes theorem using geometric elements of the barycentric dual grid, providing the equivalence between mimetic numerical schemes and discrete geometric approaches. Second, we introduce a novel global property of the reconstruction operators, called P0-consistency, which extends the standard consistency requirement of the mimetic framework. This concept characterizes the whole class of reconstruction operators that can be used to construct a global mass matrix in such a way that a global patch test is passed. Given the geometric description of the scheme, we can set up a correspondence between entries of reconstruction operators and geometric elements of a secondary grid, which is built by duality from the primary grid used in the scheme formulation. Finally, we show the that the geometric interpretation is necessary for the correct evaluation of certain physical variables in the post-processing stage. A discussion on how the geometric viewpoint allows to optimize reconstruction operators completes the exposition.

Simulation of the Laminar–Turbulent Transition by Applying Dissipative Numerical Schemes

The applicability of a low-order monotone shock-capturing scheme to the simulation of the laminar–turbulent transition is demonstrated. The laminar–turbulent transition is simulated in a supersonic boundary layer over a flat plate at a Mach number of 3. The numerical results are compared with results of other authors based on low-dissipative schemes. The spectral characteristics of disturbances in the linear and nonlinear development regions, the structure of the transient flow, and averaged boundary layer characteristics are compared.

Index modulation for OFDM RadCom systems

This paper studies a novel Radar and wireless communication fusion system namely, RadCom, using orthogonal frequency division multiplexing index modulation waveforms. From the communication perspective, information is conveyed not only through modulated orthogonal frequency division multiplexing subcarriers but also by the active subcarrier indices that are dynamically updated at the transmitted symbol rates. When compared with conventional orthogonal frequency division multiplexing communication systems, only a subset of subcarriers is activated in the proposed orthogonal frequency division multiplexing index modulation system, which reduces signal peak-to-average ratios, leading to enhanced power amplifier efficiency. From the Radar sensing perspective, the Radar range-velocity profile obtained using compressed sensing approach is simulated and presented. It is shown that orthogonal frequency division multiplexing index modulation RadCom system can enhance the transmission data rates when low-order modulation schemes are adopted e.g. binary phase shift keying, and maintain similar Radar performance compared with the orthogonal frequency division multiplexing counterpart when the system is carefully designed.

On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity

We propose and analyze two locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity. One is a high-order scheme, and the other is a low-order scheme. For time discretization, we use the backward Euler scheme. The proposed methods are well-posed, and optimal error estimates of all the unknowns are obtained for fully discrete solutions. The generic constants in the estimates are uniformly bounded as the Lamé coefficient λ tends to infinity, and as the constrained specific storage coefficient is arbitrarily small. Therefore the methods are free of both Poisson locking and pressure oscillations. Numerical results illustrate the good performance of the methods and confirm our theoretical predictions.

A Fast Volume Integral Equation Solver With Linear Basis Functions for the Accurate Computation of EM Fields in MRI

A stable volume integral equation (VIE) solver based on polarization/magnetization currents is presented, for the accurate and efficient computation of the electromagnetic (EM) scattering from highly inhomogeneous and high contrast objects. We employ the Galerkin method of moments to discretize the formulation with discontinuous piecewise linear basis functions on uniform voxelized grids, allowing for the acceleration of the associated matrix-vector products in an iterative solver, with the help of FFT. Numerical results illustrate the superior accuracy and more stable convergence properties of the proposed framework, when compared against standard low-order (piecewise constant) discretization schemes and a more conventional VIE formulation based on electric flux densities. Finally, the developed solver is applied to analyze complex geometries, including realistic human body models, typically used in modeling the interactions between EM waves and biological tissue.

Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

This article contributes to a balanced space–time spectral collocation method for solving nonlinear time‐fractional Burgers equations with given initial‐boundary conditions. Most of existing approximate methods for solving partial differential equations are unbalanced, since they have used a low order scheme such as finite difference methods for integrating the temporal variable and a high order numerical framework such as spectral Galerkin (or meshless) method for discretization of space variables. So in the current paper, our suggested scheme is balanced in both time and space variables. Due to the non‐smoothness of solutions of time‐fractional Burgers equations, we apply efficient basis functions as the fractional Lagrange functions for interpolating time variable. By collocating the main equation and the initial‐boundary conditions together with the implementation of the corresponding operational matrices of spatial and fractional temporal variables, the assumed model is transformed into the associated system of nonlinear algebraic equations, which can be solved via efficient iterative solvers such as the Levenberg–Marquardt method. Also, we fully analyze the convergence of method. Moreover, we consider several test problems for examining the suggested scheme that confirms its high accuracy and low computational cost with respect to recent numerical methods in the literature.

A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations

The main result in this paper is a provably entropy stable shock capturing approach for the high order entropy stable Discontinuous Galerkin Spectral Element Method (DGSEM) based on a hybrid blending with a subcell low order variant. Since it is possible to rewrite a high order summation-by-parts (SBP) operator into an equivalent conservative finite volume form, we were able to design a low order scheme directly with the Legendre-Gauss-Lobatto (LGL) nodes that is compatible to the discrete entropy analysis used for the proof of the entropy stable DGSEM. Furthermore, we present a hybrid low order/high order discretisation where it is possible to seamlessly blend between the two approaches, while still being provably entropy stable. With tensor products and careful design of the low order scheme on curved elements, we are able to extend the approach to three spatial dimensions on unstructured curvilinear hexahedral meshes. We validate our theoretical findings and demonstrate convergence order for smooth problems, conservation of the primary quantities and discrete entropy stability for an arbitrary blending on curvilinear grids. In practical simulations, we connect the blending factor to a local troubled element indicator that provides the control of the amount of low order dissipation injected into the high order scheme. We modified an existing shock indicator, which is based on the modal polynomial representation, to our provably stable hybrid scheme. The aim is to reduce the impact of the parameters as good as possible. We describe our indicator in detail and demonstrate its robustness in combination with the hybrid scheme, as it is possible to compute all the different test cases without changing the indicator. The test cases include e.g. the double Mach reflection setup, forward and backward facing steps with shock Mach numbers up to 100. The proposed approach is relatively straight forward to implement in an existing entropy stable DGSEM code as only modifications local to an element are necessary.

The Double Absorbing Boundary Method Incorporated in a High-Order Spectral Element Formulation

In this paper, we consider the numerical solution of the time-dependent wave equation in a semi-infinite waveguide. We employ the Double Absorbing Boundary (DAB) method, by introducing two parallel artificial boundaries on the side where waves are outgoing. In contrast to the original implementation of the DAB, where the numerical solution involved either a low-order finite difference scheme or a low-order finite element scheme, here we incorporate the DAB into a high-order spectral element formulation, which provides us with very accurate solutions of wave problems in unbounded domains. This is demonstrated by numerical experiments. While the method is highly accurate, it suffers from long-time instability. We show how to postpone the onset of the instability by a prudent choice of the computational parameters.

Spectral collocation in space and time for linear PDEs

Spectral methods solve elliptic partial differential equations (PDEs) numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods that converge spectrally in both space and time have appeared recently. This paper is a continuation of the authors' previous works on Legendre and Chebyshev space-time methods for the heat equation. Here space-time spectral collocation methods for the Schrodinger, wave, Airy and beam equations are proposed and analyzed. In particular, a condition number estimate of each global Chebyshev space-time operator is shown. The analysis requires new estimates of eigenvalues of some spectral derivative matrices. In particular, it is shown that the real part of every eigenvalue of the third-order Chebyshev derivative matrix is positive and bounded away from zero, settling a twenty-year-old conjecture. Similarly, the real part of every eigenvalue of the fourth-order Chebyshev derivative matrix with Dirichlet boundary conditions is shown to be also positive and bounded away from zero. Numerical results verify the theoretical results, and demonstrate that the space-time methods also work well for some common nonlinear PDEs.

Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws

In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable/entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems.

Characterization of Orthogonal Frequency Division Multiplexing (OFDM) Communication Links

The demand for high data rate transmission is rapidly increasing in the area of wireless communications and Orthogonal Frequency Division Multiplexing (OFDM) is known to be a promising technique for high-rate transmission that can overcome the inter symbol interference (ISI) which results from the time dispersive nature of wireless channels. The idea is to utilize a number of carriers, spread regularly over a frequency band, in such a way so that the available bandwidth is utilized to maximal efficiency. This study presents the comparison of the performance of different modulation schemes: M-ary phase shift keying (M-PSK) and M-ary quadrature amplitude modulation (M-QAM) over Additive White Gaussian Noise (AWGN), Rayleigh and Rician multipath fading channels in OFDM system. Bit error rate (BER) was used to evaluate the performance of the system; the number of fast Fourier transform (FFT) points used during the transmission was also examined. The results showed that for all the channels, lower order modulation schemes perform better than the higher order schemes. This comes at the detriment of the data rate, as lower order schemes have lower data rates compared with their higher order counterparts. In addition, it was observed that the system performed better in AWGN channel than Rayleigh and Rician fading channels for all the modulation schemes used. It was observed that the number of FFT points used during the transmission do not have much effects on the performance of the system.

Phong deformation

Physics-based simulations of deforming tetrahedral meshes are widely used to animate detailed embedded geometry. Unfortunately most practitioners still use linear interpolation (or other low-order schemes) on tetrahedra, which can produce undesirable visual artifacts, e.g., faceting and shading artifacts, that necessitate increasing the simulation's spatial resolution and, unfortunately, cost. In this paper, we propose Phong Deformation , a simple, robust and practical vertex-based quadratic interpolation scheme that, while still only C 0 continuous like linear interpolation, greatly reduces visual artifacts for embedded geometry. The method first averages element-based linear deformation models to vertices, then barycentrically interpolates the vertex models while also averaging with the traditional linear interpolation model. The method is a fast, robust, and easily implemented replacement for linear interpolation that produces visually better results for embedded deformation with irregular tetrahedral meshes.

A posteriori error estimation for the spectral deferred correction method

The spectral deferred correction method is a variant of the deferred correction method for solving ordinary differential equations. A benefit of this method is that is uses low order schemes iteratively to produce a high order approximation. In this paper we consider adjoint-based a posteriori analysis to estimate the error in a quantity of interest of the solution. This error formula is derived by first developing a nodally equivalent finite element method to the spectral deferred correction method. The error formula is then split into various terms, each of which characterizes a different component of the error. These components may be used to determine the optimal strategy for changing the method parameters to best improve the error.