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Inspired by the methods developed in J. H. Bramble, T. Dupont, and V. Thomée (“Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections,” Math. Comput., vol. 26, no. 120, pp. 869–879, 1972), we introduce a new technique that yields a symmetric formulation and has similar performance. The new method is based on a Robin-type problem on an approximate polygonal domain. Optimal error estimates in the energy norm are proved for piecewise quadratics and cubics. We provide numerical experiments that show our theoretical results are sharp. [ABSTRACT FROM AUTHOR]

With the rapid development of the information age, the need for high-resolution reconnaissance and surveillance is becoming more and more urgent. It is necessary to develop photoelectric pods with a high-precision stabilization function, which isolate the influence of external disturbance and realize the tracking of maneuvering targets. In this paper, the internal frame stabilization loop control technique is studied. Firstly, the mathematical models of the current loop are established. Secondly, the friction model, parametric model, and mechanical resonance model of the system are identified. Finally, a fourth-order tracking differentiator and a fifth-order extended state observer are designed. Through simulation verification, the stability performance of HO-ADRC, increasing by 145.17%, is better than that of PID. In terms of disturbance suppression and noise removal ability, HO-ADRC is also better than PID. [ABSTRACT FROM AUTHOR]

In solving problems from Computational Fluid Dynamics (CFD) and physics, huge efforts should be afforded to obtain accurate and applicable schemes for the derivatives. Based on the idea of high order polynomials, many sets of second derivative schemes are derived in this paper. These sets are grouped according to the order of the accuracy of the approximations from order three to order seven. Different types of second derivative forward, central, and backward compact with some traditinal approximations are inrtoduced at each set by the proposed method. The order of accuracy is verified of each scheme using the technique of finding the values of the coefficients for the error terms by matching both sides of the given scheme. Many schemes that are introduced in this article are applicable to some problems from science and engineering. [ABSTRACT FROM AUTHOR]

This paper deals with high order Whitney forms. We define a canonical isomorphism between two sets of degrees of freedom. This allows to geometrically localize the classical degrees of freedom, the moments, over the elements of a simplicial mesh. With such a localization, it is thus possible to associate, even with moments, a graph structure relating a field with its potential. [ABSTRACT FROM AUTHOR]

In our industry, active disturbance rejection control already has been used to enhance the performance of the electro-hydraulic servo systems, despite the fact that electro-hydraulic servo systems are usually reduced to first-order and second-order systems. The aim of this paper is to extend the application of active disturbance rejection control to high-order electro-hydraulic servo systems by introducing a new tuning method. Active disturbance rejection control is transformed into two separate parts in the frequency domain: a pre-filter H (s) and a controller T (s) . The parameters of the pre-filter and controller can be tuned to satisfy the performance requirements of high-order electro-hydraulic servo systems using quantitative feedback theory. To assess the efficacy of the proposed tuning approach, simulations and an application of a third-order electro-hydraulic servo system have been carried out and the stability of the application with an improved active disturbance rejection controller is analyzed. The results of simulations and experiments reveal that the new tuning method for high-order electro-hydraulic servo systems can obtain a better performance than the bandwidth tuning method and other methods. [ABSTRACT FROM AUTHOR]

It is common for mathematical models of physical systems to possess qualitative properties such as positivity, monotonicity, or conservation of underlying physical behavior. When these models consist of differential equations, it is also common for them to be solved via splitting, i.e., splitting the differential equations into parts that are integrated separately. All splitting strategies are not created equal; however, in this work, we study the effect of two splitting strategies on qualitative property preservation applied to the basic susceptible-infected-recovered (SIR) model from epidemiology and the effect of backward integration of operator-splitting methods on positivity preservation in the Robertson test problem. We find that qualitative property preservation does depend on the splitting strategy even if the sub-integrations are performed exactly. Accordingly, the specific choice of splitting strategy used may be informed by requirements of qualitative property preservation. The choice of operator-splitting method also depends on the specific properties of the exact solution of the sub-systems. [ABSTRACT FROM AUTHOR]

In this paper, we provide high order hybrid asymptotic augmented finite volume schemes on a uniform grid for nonlinear weakly degenerate and strongly degenerate wave equations. The whole domain is divided into singular and regular subdomains by introducing an intermediate point. Puiseux series asymptotic technique is used in singular subdomain, and augmented numerical method is used in regular subdomain. The key to the method are the recovery of Puiseux series for the nonlinear degenerate wave equation in singular subdomain and the organic combination between the singular and regular subdomains by means of augmented variables related to singularity. In particular, through imposing a condition at the intermediate point, we can not only improve the accuracy of the augmented variables, but also avoid the restriction conditions when the mesh is divided in regular subdomain. The advantage of this method is that the global convergence order of the degenerate wave equation is determined by the augmented numerical scheme in regular subdomain. A rigorous error estimate is conducted for the solution of the degenerate wave equation. Numerical examples on weakly degenerate and strongly degenerate problems are provided to illustrate the effectiveness of the proposed method. Especially, we use the method to solve an interesting example of a degenerate wave equation with coefficient blow-up. [ABSTRACT FROM AUTHOR]

With the rapid development of the information age, the need for high-resolution reconnaissance and surveillance is becoming more and more urgent. It is necessary to develop photoelectric pods with a high-precision stabilization function, which isolate the influence of external disturbance and realize the tracking of maneuvering targets. In this paper, the internal frame stabilization loop control technique is studied. Firstly, the mathematical models of the current loop are established. Secondly, the friction model, parametric model, and mechanical resonance model of the system are identified. Finally, a fourth-order tracking differentiator and a fifth-order extended state observer are designed. Through simulation verification, the stability performance of HO-ADRC, increasing by 145.17%, is better than that of PID. In terms of disturbance suppression and noise removal ability, HO-ADRC is also better than PID.

The Neilan Pk$$ {P}_k $$‐Pk−1$$ {P}_{k-1} $$ divergence‐free finite element is stable on any tetrahedral grid, where the piece‐wise Pk$$ {P}_k $$ polynomial velocity is C0$$ {C}^0 $$ on the grid, C1$$ {C}^1 $$ on edges and C2$$ {C}^2 $$ at vertices, and the piece‐wise Pk−1$$ {P}_{k-1} $$ polynomial pressure is C0$$ {C}^0 $$ on edges and C1$$ {C}^1 $$ at vertices. However the method does not work if the exact pressure solution does not vanish on all domain edges, because of the excessive continuity requirements. We extend the Neilan element by removing the extra requirements at domain boundary edges. That is, if a vertex is on a domain boundary edge and if an edge has one endpoint on a domain boundary edge, the velocity is only C0$$ {C}^0 $$ at the vertex and on the edge, respectively, and the pressure is totally discontinuous there. Under the condition that no tetrahedron in the grid has more than one face‐triangle on the domain boundary, we prove that the extended finite element is stable, and consequently produces solutions of optimal order convergence for all Stokes problems. A numerical example is given, confirming the theory. [ABSTRACT FROM AUTHOR]

A fully discrete finite element method with a Gauss collocation in time is proposed for solving the nonlinear Schrödinger equation with a wave operator in the d-dimensional torus, d ∈ { 1 , 2 , 3 } . Based on Gauss collocation method in time and the scalar auxiliary variable technique, the proposed method preserves both mass and energy conservations at the discrete level. Existence and uniqueness of the numerical solutions to the nonlinear algebraic system, as well as convergence to the exact solution with order O (h p + τ k + 1) in the L ∞ (0 , T ; H 1) norm, are proved by using Schaefer’s fixed point theorem without requiring any grid-ratio conditions, where (p, k) is the degree of the space-time finite elements. The Newton iterative method is applied for solving the nonlinear algebraic system. The numerical results show that the proposed method preserves discrete mass and energy conservations up to machine precision, and requires only a few Newton iterations to achieve the desired accuracy, with optimal-order convergence in the L ∞ (0 , T ; H 1) norm. [ABSTRACT FROM AUTHOR]

In our industry, active disturbance rejection control already has been used to enhance the performance of the electro-hydraulic servo systems, despite the fact that electro-hydraulic servo systems are usually reduced to first-order and second-order systems. The aim of this paper is to extend the application of active disturbance rejection control to high-order electro-hydraulic servo systems by introducing a new tuning method. Active disturbance rejection control is transformed into two separate parts in the frequency domain: a pre-filter H(s) and a controller T(s). The parameters of the pre-filter and controller can be tuned to satisfy the performance requirements of high-order electro-hydraulic servo systems using quantitative feedback theory. To assess the efficacy of the proposed tuning approach, simulations and an application of a third-order electro-hydraulic servo system have been carried out and the stability of the application with an improved active disturbance rejection controller is analyzed. The results of simulations and experiments reveal that the new tuning method for high-order electro-hydraulic servo systems can obtain a better performance than the bandwidth tuning method and other methods.

In this manuscript, we develop and analyze two high‐order schemes, CFD g−σ$$ {}_{g-\sigma } $$ and PQS g−σ$$ {}_{g-\sigma } $$, for generalized variable coefficients fractional reaction–diffusion equations. The generalized fractional derivative is characterized by a weight function and a scale function. We approximate it using generalized Alikhanov formula (gL2−1σ$$ gL2-{1}_{\sigma } $$) of order (3−μ)$$ \left(3-\mu \right) $$, where μ$$ \mu $$(0<μ<1)$$ \left(0&amp;lt;\mu &amp;lt;1\right) $$ denotes the order of the generalized fractional derivative. Moreover, for spatial discretization, we use a compact operator in CFD g−σ$$ {}_{g-\sigma } $$ scheme and parametric quintic splines in PQS g−σ$$ {}_{g-\sigma } $$ scheme. The stability and convergence analysis of both schemes are demonstrated thoroughly using the discrete energy method in the L2$$ {L}_2 $$‐norm. It is shown that the convergence orders of the CFD g−σ$$ {}_{g-\sigma } $$ and PQS g−σ$$ {}_{g-\sigma } $$ schemes are O(Δt3−μ,Δ˜t2,h4)$$ O\left(\Delta {t}&amp;amp;#x0005E;{3-\mu },\tilde{\Delta}{t}&amp;amp;#x0005E;2,{h}&amp;amp;#x0005E;4\right) $$ and O(Δt3−μ,Δ˜t2,h4.5)$$ O\left(\Delta {t}&amp;amp;#x0005E;{3-\mu },\tilde{\Delta}{t}&amp;amp;#x0005E;2,{h}&amp;amp;#x0005E;{4.5}\right) $$, respectively, where Δt$$ \Delta t $$ and Δ˜t$$ \tilde{\Delta}t $$ represent the mesh spacing in the time direction and h$$ h $$ is the mesh spacing in the space direction. In addition, numerical results are obtained for three test problems to validate the theory and demonstrate the efficiency and superiority of the proposed schemes. [ABSTRACT FROM AUTHOR]

We present hipBone, an open-source performance-portable proxy application for the Nek5000 (and NekRS) computational fluid dynamics applications. HipBone is a fully GPU-accelerated C++ implementation of the original NekBone CPU proxy application with several novel algorithmic and implementation improvements which optimize its performance on modern fine-grain parallel GPU accelerators. Our optimizations include a conversion to store the degrees of freedom of the problem in assembled form in order to reduce the amount of data moved during the main iteration and a portable implementation of the main Poisson operator kernel. We demonstrate near-roofline performance of the operator kernel on three different modern GPU accelerators from two different vendors. We present a novel algorithm for splitting the application of the Poisson operator on GPUs which aggressively hides MPI communication required for both halo exchange and assembly. Our implementation of nearest-neighbor MPI communication then leverages several different routing algorithms and GPU-Direct RDMA capabilities, when available, which improves scalability of the benchmark. We demonstrate the performance of hipBone on three different clusters housed at Oak Ridge National Laboratory, namely, the Summit supercomputer and the Frontier early-access clusters, Spock and Crusher. Our tests demonstrate both portability across different clusters and very good scaling efficiency, especially on large problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonautonomous high-order delay differential equation with 2 r − 1 lags. The 4 r -periodic orbits are obtained by using the variational method and a new Z n index theory. This is a new type of nonautonomous delay differential equation compared with all existing ones. An example is given to demonstrate our main results. [ABSTRACT FROM AUTHOR]

Imposition methods of interface conditions for the second-order wave equation with non-conforming grids is considered. The spatial discretization is based on high order finite differences with summation-by-parts properties. Previously presented solution methods for this problem, based on the simultaneous approximation term (SAT) method, have shown to introduce significant stiffness. This can lead to highly inefficient schemes. Here, two new methods of imposing the interface conditions to avoid the stiffness problems are presented: (1) a projection method and (2) a hybrid between the projection method and the SAT method. Numerical experiments are performed using traditional and order-preserving interpolation operators. Both of the novel methods retain the accuracy and convergence behavior of the previously developed SAT method but are significantly less stiff. [ABSTRACT FROM AUTHOR]

When solving numerically an elliptic problem, it is important in most applications that the scheme used preserves the positivity of the solution. When using finite volume schemes on deformed meshes, the question has been solved rather recently. Such schemes are usually (at most) second-order convergent, and non-linear. On the other hand, many high-order schemes have been proposed that do not ensure positivity of the solution. In this paper, we propose a very high-order monotonic (that is, positivity preserving) numerical method for elliptic problems in 1D. We prove that this method converges to an arbitrary order (under reasonable assumptions on the mesh) and is indeed monotonic. We also show how to handle discontinuous sources or diffusion coefficients, while keeping the order of convergence. We assess the new scheme, on several test problems, with arbitrary (regular, distorted, and random) meshes. [ABSTRACT FROM AUTHOR]

In this paper, we study a geometric approach for constructing physical degrees of freedom for sequences of finite element spaces. Within the framework of finite element systems, we propose new degrees of freedom for the spaces P r Λ k of polynomial differential forms and we verify numerically their unisolvence. [ABSTRACT FROM AUTHOR]

In this work, we construct and analyze three dimensional high order finite volume element schemes for elliptic equations. In these schemes, the trial function space is chosen as the standard rth order Lagrange finite element space, where r≥1$$ r\ge 1 $$ is an arbitrary positive integer; the test function space is chosen as the piecewise constant space with respect to the dual mesh of which the control volumes are constructed using Gauss points in each element of the primal mesh. We investigate the inf–sup property of these schemes and based on it, we prove that u−uh1=Ohr$$ {\left|u-{u}_h\right|}_1=\mathcal{O}\left({h}^r\right) $$, uI−uh1=Ohr+1$$ {\left|{u}_I-{u}_h\right|}_1=\mathcal{O}\left({h}^{r+1}\right) $$ and ‖u−uh‖0=Ohr+1$$ {\left\Vert u-{u}_h\right\Vert}_0=\mathcal{O}\left({h}^{r+1}\right) $$, where u$$ u $$ is the exact solution, uh$$ {u}_h $$ is the rth order finite volume element solution and uI$$ {u}_I $$ is the piecewise rth order Lagrange interpolation of u$$ u $$. Several numerical examples are presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this study, the numerical manifold method (NMM), coupled with material interpolation, is applied to the topology optimization of continuum structures. Quadrilateral element (Q4) meshes are used in the NMM for mathematical cover. The order of the local approximation function is increased to improve the accuracy of the analysis. This method usually leads to linear dependency of the global degrees of freedom, as reflected in the rank deficiency of the global approximation space. However, using Q4 elements, the rank deficiency value is not fixed with mesh refinement. The presented solution handles this problem, and the efficiency of the proposed method is investigated with four examples. The results indicate that the high-order numerical manifold method (HONMM) has great accuracy in the analysis and works well for topology optimization problems. One advantage of using the combined HONMM with material interpolation in topology problems is chequerboard-free patterns, owing to mathematical cover and high-order analysis. [ABSTRACT FROM AUTHOR]

In this article, one dimensional non-linear sine-Gordon equation has been studied. We propose a new method based on collocation of cubic B spline to find the numerical solution of non-linear sine-Gordon equation with Dirichlet boundary conditions. The method involves high order perturbations of the classical cubic spline collocation method at the partition nodes. The existence and uniqueness of the numerical solution has been proved. The method produces optimal fourth order accurate solutions. Stability analysis of the method has been done. The numerical experiments confirm the expected accuracy. We compare the results obtained by our method with those by other techniques for some already existing examples in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a new type of weighted essentially non-oscillatory scheme with variable index (VWENO) is obtained. The index can adaptively adjust with the solution, to ensure that the VWENO scheme uses optimal weights in smooth regions, while non-linear weights are used in less smooth regions. Theoretical and numerical results show that the variable index can make the result of VWENO achieve the optimal weights in the smooth regions without amplifying the weight of less smooth sub-stencils containing discontinuities. Theoretical and numerical calculation experiments show that the new scheme's shock capture capability and the resolution of complex process structures are significantly better than WENO-JS and WENO-Z. [ABSTRACT FROM AUTHOR]

The three-temperature (3-T) radiation hydrodynamics (RH) equations play an important role in the high-energy-density-physics fields, such as astrophysics and inertial confinement fusion (ICF). It describes the interaction between radiation and high-energy-density plasmas including electron and ion in the assumption that radiation, electron and ion are in their own equilibrium state, which means they can be characterized by their own temperatures. The 3-T RH system consists of the density, momentum and three internal energy (electron, ion and radiation) equations. In this paper, we propose a high order conservative finite difference weighted essentially non-oscillatory (WENO) scheme solving one-dimensional (1D) and two-dimensional (2D) 3-T RH equations respectively. Following our previous paper [7] , we introduce the three new energy variables, and then design a finite difference scheme with both the conservative property and arbitrary high order accuracy. Based on the WENO interpolation and the strong stability preserving (SSP) high order time discretizations, taken as an example, we design a class of fifth order conservative finite difference schemes in space and third order in time. Compared with the Lagrangian method we proposed in [7] , which can only reach second order accuracy for 2D 3-T RH equations if straight-line edged meshes are used, the finite difference scheme can be easily designed to arbitrary high order accuracy for multi-dimensional 3-T RH equations. The finite difference formulation is also much less expensive in multi-dimensions than finite volume schemes used in [7]. Furthermore, our method can handle fluids with large deformation easily. Numerous 1D and 2D numerical examples are presented to verify the desired properties of the high order finite difference WENO schemes such as high order accuracy, non-oscillation, conservation and adaptation to severely distorted single-material radiation hydrodynamics problems. [ABSTRACT FROM AUTHOR]

Abstract A high fidelity flow simulation for complex geometries for high Reynolds number (Re) flow is still very challenging, requiring a more powerful HPC system. However, the development of HPC with traditional CPU architecture suffers bottlenecks due to its high power consumption and technical difficulties. Heterogeneous architecture computation is raised to be a promising solution to the challenges of HPC development. GPU accelerating technology has been utilized in low order scheme CFD solvers on the structured grid and high order scheme solvers on unstructured meshes. The high-order finite difference methods on structured grids possess many advantages, e.g., high efficiency, robustness, and low storage. However, the strong dependence among points for a high-order finite difference scheme still limits its application on the GPU platform. In the present work, we propose a set of hardware-aware technology to optimize data transfer efficiency between CPU and GPU, as well as communication efficiency among GPUs. An in-house multi-block structured CFD solver with high order finite difference methods on curvilinear coordinates is ported onto the GPU platform and obtains satisfying performance with a speedup maximum of around 2000x over a single CPU core. This work provides an efficient solution to apply GPU computing in CFD simulation with specific high order finite difference methods on current GPU heterogeneous computers. The test shows that significant accelerating effects can be achieved for different GPUs.

Simulating the flow of water in district heating networks requires numerical methods which are independent of the CFL condition. We develop a high order scheme for networks of advection equations allowing large time steps. With the MOOD technique, unphysical oscillations of nonsmooth solutions are avoided. In numerical tests, the applicability to real networks is shown. [ABSTRACT FROM AUTHOR]

High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.

• The study proposed a method to predict the complex subharmonic vibrations. • Local subharmonic vibrations of order-1/3 and −1/5 were obtained accurately. • Local "full tree" and "half tree" of subharmonic vibrations were predicted. • Strange vibrations in the real turbo-engine testings were explained. • Experiments were conducted for comparing with the theoretical predictions. In real turbomachinery testing, we always confront strange high order vibrations which scatter a few points or draw some local lines in the experimental bifurcation diagrams. Such vibrations maybe ignored due to the lack of experiences and knowledge while they can be important in design and operation due to the nonsmooth jumping and instability. This paper proposes a new method for predicting the complex high order subharmonic vibrations for a nonlinear rotor system with light rub. The nonlinearity is induced by the contact of the bristles and the rotor. A continuous nonlinear rotor system is established based on the geometric and bending force diagrams. We discretize the complex orbits and built implicit mappings between the adjacent nodes. A directional eigenfunctional matrix is constructed which maps each nodes and passes the perturbation along the orbits for semi-analytical prediction of the high order subharmonic vibrations. Stability criterion is defined. Numerical bifurcation diagrams and harmonic contours are presented for quick glance of the global subahrmonic vibrations. Two local subharmonic vibrations of odd orders are succesfuly predicted. Local high order subharmonic vibrations of "full tree" and "half tree" are predicted and achieved accurately. An experimental platform is established to compare the semi-analytical predictions with experimental results. The research delivers a new perspective for predicting and computing the high order subharmonic vibrations in nonlinear rotor system. [Display omitted] [ABSTRACT FROM AUTHOR]

Generalized isogeometric analysis (GIGA) enables accurate numerical solutions for interface problem with optimal convergence rates. However, the approach is not stable in terms of the spline functions' degree and is not robust with the position of the interface. In this paper, we introduce a stable GIGA (SGIGA) through a specific modification of the enrichment space. We prove that SGIGA can retain optimal convergence rates, and provide necessary well-conditioning for arbitrary degree and arbitrary location of the interface. We validate the theoretical results in various illustrations and compare the method with the stable generalized finite element method and GIGA. [ABSTRACT FROM AUTHOR]

• The improved HOMM is introduce dintoreliability analysis by the combination of Latin hypercube sampling and the Maximum Entropy Model. The error in failure probability is controlled by iteratively increasing the number of samples. • The time-dependent seismic reliability analysis frame work of structural level is recommended based on the improved HOMM. It directly targets the objective reliability index of the global limit states of the structure in the life cycle. • For the cold-formed steel structures subjected to prolonged exposure to the corrosive environment, the investigation indicates that the differences between time-dependent and time-independent seismic reliability indexes are substantial. Cold-formed steel structures are particularly sensitive to seismic loads due to their relatively low ductility and limited energy dissipation capacity. Moreover, the small thickness of cold-formed steel structures makes them more susceptible to environmental influences, leading to corrosion and material degradation. Therefore, it is essential to quantitatively assess the seismic reliability of cold-formed steel structures from a lifecycle perspective. Current structural reliability design standards often focus on the ultimate strength at the component level while neglecting the global safety of structures. Additionally, it tends to neglect the time-variant characteristic of structural resistance due to the corrosion and degradation of materials. Therefore, the time-dependent seismic reliability analysis framework was proposed at the global structural level based on an improved high-order moment method (HOMM). It directly addresses the objective failure probabilities at the global structural level and takes into account the resistance degradation of structures in the design reference period. A comparative analysis of the time-dependent seismic reliability between rocking cold-formed steel frames and rigid cold-formed steel frames is conducted, focusing on load-bearing capacity limit states and deformation capacity limit states. The investigation indicates that the differences between time-dependent and time-independent seismic reliability indexes are substantial. For cold-formed steel structures subjected to prolonged exposure to the corrosive environment, it is recommended to consider the time-dependent degradation of structures. [Display omitted] [ABSTRACT FROM AUTHOR]

The WENO-NIP scheme was obtained by developing a class of L 1 -norm smoothness indicators based on Newton interpolation polynomial. It recovers the optimal convergence order in smooth regions regardless of critical points and achieves better resolution than the classical WENO-JS scheme. However, the WENO-NIP scheme produces severe spurious oscillations when solving 1D linear advection problems with discontinuities at long output times, and it is also very oscillatory near discontinuities for 1D Riemann problems. In this paper, we find that the spectral property of WENO-NIP exhibits the negative dissipation characteristic, and this is the reason why WENO-NIP is unstable near discontinuities. Using this knowledge, we develop a way of improving the WENO-NIP scheme by introducing an additional term to eliminate the negative dissipation interval. The proposed scheme, denoted as WENO-NIP+, maintains the same convergence property, as well as the same low-dissipation property, as the corresponding WENO-NIP scheme. Numerical examples confirm that the proposed scheme is much more stable near discontinuities for 1D linear advection problems with large output times and 1D Riemann problems than the WENO-NIP scheme. Furthermore, the new scheme is far less dissipative in the region with high-frequency waves. In addition, the improved WENO-NIP+ scheme can remove or at least greatly decrease the post-shock oscillations that are commonly produced by the WENO-NIP scheme when simulating 2D Euler equations with strong shocks. [ABSTRACT FROM AUTHOR]

This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge–Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, we set suitable definitions of consistency and stability for these methods. This allows for a proof that arbitrarily high order linearly implicit methods exist and converge when applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs that illustrate our theoretical results for ODEs, and compare our methods with standard methods for several evolution PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following high order degenerate elliptic system: $\left\{ \begin{align} & {{(-A)}^{m}}u={{v}^{p}} \\ & {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y>0 \right\}, \\ & u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(1 \right)$ [ABSTRACT FROM AUTHOR]

We propose high-order FDTD schemes based on the Correction Function Method (CFM) (Marques et al. in J Comput Phys 230:7567–7597, 2011) for Maxwell’s interface problems with discontinuous coefficients and complex interfaces. The key idea of the CFM is to model the correction function near an interface to retain the order of a finite difference approximation. To do so, we solve a system of PDEs based on the original problem by minimizing an energy functional. The CFM is applied to the standard Yee scheme and a fourth-order FDTD scheme. The proposed CFM-FDTD schemes are verified in 2-D using the transverse magnetic ( TM z ) mode. Numerical examples include scattering of magnetic and non-magnetic dielectrics, and problems with manufactured solutions using various complex interfaces and discontinuous piecewise varying coefficients. Long-time simulations are also performed to investigate the stability of CFM-FDTD schemes. The proposed CFM-FDTD schemes achieve up to fourth-order convergence in L 2 -norm and provide approximations devoid of spurious oscillations. [ABSTRACT FROM AUTHOR]

An exponential type of convolution quadrature is proposed as a time-stepping method for the nonlinear subdiffusion equation with bounded measurable initial data. The method combines contour integral representation of the solution, quadrature approximation of contour integrals, multistep exponential integrators for ordinary differential equations, and locally refined stepsizes to resolve the initial singularity. The proposed k-step exponential convolution quadrature can have kth-order convergence for bounded measurable solutions of the nonlinear subdiffusion equation based on naturalregularity of the solution with bounded measurable initial data. [ABSTRACT FROM AUTHOR]

Annotation: The mixed problem for a degenerate high order equation with a fractional derivative in a rectangular domain is considered in the article. The existence of a solution and its uniqueness are shown by the spectral method. [ABSTRACT FROM AUTHOR]

The current research paper discusses the implementation of higher order-matched filter design using odd and even phase processes for efficient area and time delay reduction. Matched filters are widely used tools in the recognition of specified task. When higher order taps are implemented upon the transposed form of matched filters, it can enhance the image recognition application and its performance in terms of identification and accuracy. The proposed method i.e., odd and even phases’ process of FIR filter can reduce the number of multipliers and adders, used in existing system. The main advantage of using higher order tap-matched filter is that it can reduce the area required, owing to its odd and even processes. Further, it also successfully reduces the time delay, especially in case of high order demands. The performance of higher order matched filter design, using odd and even phase process, was analyzed using Xilinx 9.1 ISE Simulator. The study results accomplished reduction in area, 70% increase in throughput compared to traditional implementation and reduced time delay. In addition to these, Vedic multiplier-based FIR is modified with a tree-based MAM that reduces the number of shifter and adder to replace the multiplier. [ABSTRACT FROM AUTHOR]

We present a Hermite interpolation based partial differential equation solver for Hamilton–Jacobi equations. Many Hamilton–Jacobi equations have a nonlinear dependency on the gradient, which gives rise to discontinuities in the derivatives of the solution, resulting in kinks. We built our solver with two goals in mind: (1) high order accuracy in smooth regions and (2) sharp resolution of kinks. To achieve this, we use Hermite interpolation with a smoothness sensor. The degrees-of-freedom of Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate direction. The method uses (m + 1) d degrees of freedom per node in d-dimensions and achieves an order of accuracy (2 m + 1) when the solution is smooth. To obtain sharp resolution of kinks, we sense the smoothness of the solution on each cell at each timestep. If the solution is smooth, we march the interpolant forward in time with no modifications. When our method encounters a cell over which the solution is not smooth, it introduces artificial viscosity locally while proceeding normally in smooth regions. We show through numerical experiments that the solver sharply captures kinks once the solution losses continuity in the derivative while achieving 2 m + 1 order accuracy in smooth regions. [ABSTRACT FROM AUTHOR]

A high fidelity flow simulation for complex geometries for high Reynolds number (Re) flow is still very challenging, requiring a more powerful HPC system. However, the development of HPC with traditional CPU architecture suffers bottlenecks due to its high power consumption and technical difficulties. Heterogeneous architecture computation is raised to be a promising solution to the challenges of HPC development. GPU accelerating technology has been utilized in low order scheme CFD solvers on the structured grid and high order scheme solvers on unstructured meshes. The high-order finite difference methods on structured grids possess many advantages, e.g., high efficiency, robustness, and low storage. However, the strong dependence among points for a high-order finite difference scheme still limits its application on the GPU platform. In the present work, we propose a set of hardware-aware technology to optimize data transfer efficiency between CPU and GPU, as well as communication efficiency among GPUs. An in-house multi-block structured CFD solver with high order finite difference methods on curvilinear coordinates is ported onto the GPU platform and obtains satisfying performance with a speedup maximum of around 2000x over a single CPU core. This work provides an efficient solution to apply GPU computing in CFD simulation with specific high order finite difference methods on current GPU heterogeneous computers. The test shows that significant accelerating effects can be achieved for different GPUs. [ABSTRACT FROM AUTHOR]

The susceptible-infected-recovered (SIR) model is perhaps the most basic epidemiological model for the evolution of disease spread within a population. Because of its direct representation of fundamental physical quantities, a true solution to an SIR model possesses a number of qualitative properties, such as conservation of the total population or positivity or monotonicity of its constituent populations, that may only be guaranteed to hold numerically under step-size restrictions on the solver. Operator-splitting methods with order greater than two require backward sub-steps in each operator, and the effects of these backward sub-steps on the step-size restrictions for guarantees of qualitative correctness of numerical solutions are not well studied. In this study, we analyze the impact of backward steps on step-size restrictions for guaranteed qualitative properties by applying third- and fourth-order operator-splitting methods to the SIR epidemic model. We find that it is possible to provide step-size restrictions that guarantee qualitative property preservation of the numerical solution despite the negative sub-steps, but care must be taken in the choice of the method. Results such as this open the door for the design and application of high-order operator-splitting methods to other mathematical models in general for which qualitative property preservation is important. [ABSTRACT FROM AUTHOR]

We consider Galerkin schemes with discontinuous functions constructed on bases with Legendre polynomials of degree K = 2 and 3. The schemes are written to solve the one-dimensional Hopf equation. The unsteady solution is obtained using the ADER and Runge–Kutta algorithms. The declared high order of accuracy of the numerical approaches is confirmed. The computational efficiency of the ADER method is investigated in comparison with the traditional approach. Test that have an analytical solution (linear solution and a traveling half-wave) and a task with Burgers turbulence are used as tests. The results of this study can be used to accelerate three-dimensional algorithms based on the Galerkin scheme. [ABSTRACT FROM AUTHOR]