Higher-order Approximations Research Articles

A kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator with higher approximation order

In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement.

Modelling of the porous medium flow with pressure-dependent viscosity and drag coefficient

Abstract In this paper we consider the porous medium flow through a corrugated channel where the viscosity of the fluid can significantly change with the pressure. Assuming the exponential dependence of the viscosity and drag coefficient on the pressure, we propose the higher-order approximation of the solution to the nonlinear boundary-value problem governing the flow. To accomplish that, we combine the concept of the transformed pressure with asymptotic techniques.

Invariant Domain Preserving High-Order Spectral Discontinuous Approximations of Hyperbolic Systems

Invariant Domain Preserving High-Order Spectral Discontinuous Approximations of Hyperbolic Systems

A Comparative Study of Higher Order Kernel Estimation and Kernel Density Derivative Estimation of the Gaussian Kernel Estimator with Data Application

Higher-order kernel estimation and kernel density derivative estimation are techniques for reducing the asymptotic mean integrated squared error in nonparametric kernel density estimation. A reduction in the error criterion is an indication of better performance. The estimation of kernel function relies greatly on bandwidth and the identified reduction methods in the literature are bandwidths reliant for their implementation. This study examines the performance of higher order kernel estimation and kernel density derivatives estimation techniques with reference to the Gaussian kernel estimator owing to its wide applicability in real-life-settings. The explicit expressions for the bandwidth selectors of the two techniques in relation to the Gaussian kernel and the bandwidths were accurately obtained. Empirical results using two data sets obviously revealed that kernel density derivative estimation outperformed the higher order kernel estimation excellently well with the asymptotic mean integrated squared error as the criterion function.

Spherical Harmonics for the 1D Radiative Transfer Equation. I. Reflected Light

A significant challenge in radiative transfer theory for atmospheres of exoplanets and brown dwarfs is the derivation of computationally efficient methods that have adequate fidelity to more precise, numerically demanding solutions. In this work, we extend the capability of the first open-source radiative transfer model for computing the reflected light of exoplanets at any phase geometry, PICASO (Planetary Intensity Code for Atmospheric Spectroscopy Observations). Until now, PICASO has implemented two-stream approaches to the solving the radiative transfer equation for reflected light, in particular following the derivations of Toon et al. In order to improve the model accuracy, we have considered higher-order approximations of the phase functions; namely, we have increased the order of approximation from two to four, using spherical harmonics. The spherical harmonics approximation decouples spatial and directional dependencies by expanding the intensity and phase function into a series of spherical harmonics, or Legendre polynomials, allowing for analytical solutions for low-order approximations to optimize computational efficiency. We rigorously derive the spherical harmonics method for reflected light and benchmark the four-term method (SH4) against Toon et al. and two independent and higher-fidelity methods (CDISORT and doubling method). On average, the SH4 method provides an order-of-magnitude increase in accuracy, compared to Toon et al. Finally, we implement SH4 within PICASO and observe only a modest increase in computational time, compared to two-stream methods (20% increase).

Strong form mesh-free hp-adaptive solution of linear elasticity problem

We present an algorithm for hp-adaptive collocation-based mesh-free numerical analysis of partial differential equations. Our solution procedure follows a well-established iterative solve–estimate–mark–refine paradigm. The solve phase relies on the Radial Basis Function-generated Finite Differences (RBF-FD) using point clouds generated by advancing front node positioning algorithm that supports variable node density. In the estimate phase, we introduce an Implicit-Explicit (IMEX) error indicator, which assumes that the error relates to the difference between the implicitly obtained solution (from the solve phase) and a local explicit re-evaluation of the PDE at hand using a higher order approximation. Based on the IMEX error indicator, the modified Texas Three Step marking strategy is used to mark the computational nodes for h-, p- or hp-(de-)refinement. Finally, in the refine phase, nodes are repositioned and the order of the method is locally redefined using the variable order of the augmenting monomials according to the instructions from the mark phase. The performance of the introduced hp-adaptive method is first investigated on a two-dimensional Peak problem and further applied to two- and three-dimensional contact problems. We show that the proposed IMEX error indicator adequately captures the global behaviour of the error in all cases considered and that the proposed hp-adaptive solution procedure significantly outperforms the non-adaptive approach. The proposed hp-adaptive method stands for another important step towards a fully autonomous numerical method capable of solving complex problems in realistic geometries without the need for user intervention.

Reducing the numerical dispersion of the one-way Helmholtz equation via the differential evolution method

This study is devoted to increasing the performance of the numerical methods for solving the one-way Helmholtz equation in large-scale domains. The higher-order rational approximation of the propagation operator was taken as a basis. Computation of an appropriate approximation coefficients and grid sizes is formulated as the problem of minimizing the discrete dispersion relation error. Keeping in mind the complexity of the developed optimization problem, the differential evolution method was used to tackle it. The proposed approach does not require manual selection of the artificial parameters of the numerical scheme. The stability of the scheme is provided by an additional constraint on the optimization problem. Various configurations of the differential evolution method are studied as applied to the considered problem. The advantages of the proposed method has been demonstrated on several classical diffraction problems. Comparisons with the Padé approximation method, rational interpolation and split-step Fourier method are carried out. Theoretical considerations and numerical experiments prove that the proposed method requires much less computational resources than the existing ones. Python 3 implementation of the proposed method is freely available. This work is an extended version of the ICCS-2022 conference paper (Lytaev, 2022).

A weak form quadrature element formulation of geometrically exact strain gradient shells

A geometrically exact strain gradient shell model and its weak form quadrature element formulation are proposed. The governing equations, boundary conditions and corresponding variational form of the shell model are presented. The generalized differential quadrature analogue is employed for derivative approximations of displacements and rotations with C1 continuity conditions brought about by the second-order strain gradient terms. A novel scheme based on auxiliary vectors is developed to represent and update the derivatives of constrained rotations on element edges for the numerical approach. The present quadrature shell element is feasible to incorporate strain gradient terms and free from shear and membrane locking problems due to its adjustable high-order approximation property in nonlinear shell analysis. To demonstrate the potential of this formulation for nonlinear analysis of shells with size effects, five benchmark numerical examples are presented. The results verify the feasibility of this formulation and illustrate the strain gradient effects on elastic shells undergoing large displacements and rotations.

Higher order estimate near the boundary of a large solution to semilinear Poisson equation with double-power like nonlinearity

We consider a large solution for the semilinear Poisson equation Δu=up+φ(u) in a bounded smooth domain D⊂Rn where p>1 and φ(u) behaves like αuq with q<p. We obtain the precise asymptotic behavior of a solution u near the boundary ∂D up to the third term.

An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients

Under simply supported plate (SSP) boundary conditions, a numerical method based on the higher-order Legendre polynomial approximation was studied and developed for fourth-order problems with variable coefficients. We first divide the SSP boundary conditions into two types, namely, forced boundary conditions and natural boundary conditions. According to the forced boundary conditions, an appropriate Sobolev space is defined, and a variational formulation and a discrete scheme associated with the original problem are established. Then, the existence and uniqueness of this weak solution and approximate solution are proved. By using the Céa lemma and the tensor Jacobian polynomial approximation, we further obtain the error estimation for the numerical solutions. In addition, we use the orthogonality of Legendre polynomials to construct a set of effective basis functions and derive the equivalent tensor product linear system associated with the discrete scheme, respectively. Finally, some numerical tests were carried out to validate our algorithm and theoretical analysis.

High-order conservative schemes for the nonlinear Schrödinger equation in the semiclassical limit

This letter devotes to the design of efficient prediction–correction numerical methods which produce high-order approximations of the solutions while preserving mass, or energy, or both of them, for the semiclassical Schrödinger equation with small Planck constant ɛ. The prediction step involves an explicit temporal fourth-order exponential Runge–Kutta method which allows the ɛ-oscillatory solution to be captured efficiently. The correction step only requires solving algebraic nonlinear equations. Numerical results show that the present methods have good meshing strategies τ=O(ɛ) and h=O(ɛ) and excellent power in the simulation of Bose–Einstein condensation.

Slowly rotating Kerr metric derived from the Einstein equations in affine-null coordinates

Using a quasispherical approximation of an affine-null metric adapted to an asymptotic Bondi inertial frame, we present high order approximations of the metric functions in terms of the specific angular momentum for a slowly rotating stationary and axisymmetric vacuum spacetime. The metric is obtained by following the procedure of integrating the hierarchy of Einstein equations in a characteristic formulation utilizing master functions for the perturbations. It further verified its equivalence with the Kerr metric in the slowly rotating approximation by carrying out an explicit transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates. A peculiar feature of the derivation is that in the solution of the perturbation equations for every order a new integration constant appears which cannot be set to zero using asymptotical flatness or regularity arguments. However, these additional integration constants can be absorbed into the overall Komar mass and Komar angular momentum of a slowly rotating black hole.

Dynamic analysis of bio-inspired helicoid laminated composite plates resting on Pasternak foundation excited by explosive loading

This paper uses isogeometric analysis (IGA) based on higher-order shear deformation theory (HSDT) to study the dynamic response of bio-inspired helicoid laminated composite (B-iHLC) plates resting on Pasternak foundation (PF) excited by explosive loading. IGA takes advantage of non-uniform rational B-spline (NURBS) basic functions to exactly represent the structure geometry models and the attainment of higher-order approximation conditions. This method also ensures a C1 continuous function in the analysis of transverse shear deformation via HSDT. Furthermore, IGA eliminates the requirement for correction factors and delivers accurate results. Pasternak foundation with two stiffness parameters: springer stiffness (k1) and shear stiffness (k2). The derivation of the governing equations is based on Hamilton's principle. The proposed method is validated through numerical examples. A comprehensive analysis of the impact of geometrical parameters, material properties, boundary conditions (BCs), and foundation stiffness on dynamic response of B-iHLC plates is carried out.

Finite-Element Method for the Simulation of Lipid Vesicle/Fluid Interactions in a Quasi–Newtonian Fluid Flow

We present a computational framework for modeling an inextensible single vesicle driven by the Helfrich force in an incompressible, non-Newtonian extracellular Carreau fluid. The vesicle membrane is captured with a level set strategy. The local inextensibility constraint is relaxed by introducing a penalty which allows computational savings and facilitates implementation. A high-order Galerkin finite element approximation allows accurate calculations of the membrane force with high-order derivatives. The time discretization is based on the double composition of the one-step backward Euler scheme, while the time step size is flexibly controlled using a time integration error estimation. Numerical examples are presented with particular attention paid to the validation and assessment of the model’s relevance in terms of physiological significance. Optimal convergence rates of the time discretization are obtained.

The Rochberg garden

In 1996, it was published the seminal work of Rochberg “Higher order estimates in complex interpolation theory” (Rochberg, 1996). Among many other things, the paper contains a new method to construct new Banach spaces having an intriguing behaviour: they are simultaneously interpolation spaces and twisted sums of increasing complexity. The fundamental idea of Rochberg is to consider for each z∈S the space formed by the arrays of the truncated sequence of the Taylor coefficients of the elements of the Calderón space. What was probably unforeseen is that the Rochberg constructions would lead to a deep theory connecting Interpolation theory, Homology, Operator Theory and the Geometry of Banach spaces. This work aims to synthetically present such connections, an up-to-date account of the theory and a list of significative open problems.

Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations

Abstract The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs), to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in Du et al. (2018, Commun. Comput. Phys., 24, 1049–1072) and Du et al. (2018, SIAM J. Sci. Comp., 40, A1430–A1445) to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green’s first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

Fourth order compact scheme for the Navier–Stokes equations on time deformable domains

In this work, we report the development of a spatially fourth order temporally second order compact scheme for incompressible Navier–Stokes (N–S) equations in time-varying domain. Sen (2013) put forward an implicit compact finite difference scheme for the unsteady convection–diffusion equation. It is now further extended to simulate fluid flow problems on deformable surfaces using curvilinear moving grids. The formulation is conceptualized in conjunction with recent advances in numerical grid deformations techniques such as inverse distance weighting (IDW) interpolation and its hybrid implementation. Adequate emphasis is provided to approximate grid metrics up to the desired level of accuracy and freestream preserving property has been numerically examined. As we discretize the non-conservative form of the N–S equation, the importance of accurate satisfaction of geometric conservation law (GCL) is investigated. To the best of our knowledge, this is the first higher order compact method that can directly tackle the non-conservative form of the N–S equation in single and multi-block time dependent complex regions. Several numerical verification and validation studies are carried out to illustrate the flexibility of the approach to handle high-order approximations on evolving geometries. The method presented here is found to be effective in modeling incompressible flow on deformable domains, including situations of fluid–structure interaction (FSI). Additionally, the necessity of GCL preserving metric computation is eased as a result of the scheme’s adaptation for non-conservative formulation.

Higher-order approximations in the averaging principle of multiscale systems

We study the asymptotic behavior for multiscale dynamical systems. Using the Poisson equation with parameters, we derive a hierarchy of approximation equations which are able to approximate the slow motion in the Lp-sense and achieve order ɛk/2 with any p⩾1 and k∈N+. This generalizes the averaged equation prescribed by the averaging principle, which results in order ɛ1/2 approximation.

Third-order isogeometric analysis for vibration characteristics of FGP plates in the thermal environment supported by Kerr foundation

This paper extends isogeometric analysis (IGA) based on third-order shear deformation theory (TSDT) for vibration analysis of functionally graded porous (FGP) plates in the thermal environment resting on Kerr foundation. IGA is based on non-uniform rational B-spline (NURBS) basic functions, so it can represent exact geometry and satisfy higher-order approximation (C1 continuous function) conditions of TSDT. This combination ensures high accuracy, including transverse shear deformation, without having to use any correction factors. The temperature field via the plate thickness satisfies the steady‐state heat transfer equation. The governing equations are derived from Hamilton's principle. Then, numerical examples are conducted to validate the effectiveness of the proposed method. Finally, the effects of geometrical parameters, material properties, boundary conditions (BCs), temperature change and foundation stiffness on the thermal vibration of FGP plates are fully considered.

Research on Input-Parallel Single-Switch WPT System With Load-Independent Constant Voltage Output

Single-switch circuit (SSC) in wireless power transfer (WPT) technology shows excellent performance with its simple topology. However, in higher power applications, limitation of power capacity has become the main bottleneck for SSC-WPT technology. Therefore, a research of multiple SSCs connected in parallel in WPT systems has received more attention. In this article, an input-parallel single-switch circuit (IPSSC) based on a compact decoupled coupler is proposed in WPT systems to upgrade the power capacity. Meanwhile, parallel-LCL (P-LCL) compensated network is applied to realize constant voltage output (CVO) independent of the load. Moreover, a novel method of parameter calculation using high-order harmonics approximation is proposed for IPSSC-WPT systems, which can realize precise margin of zero-voltage switching (ZVS). The experimental results of a 1-kW prototype with an output voltage of 150 V are in agreement with the theoretical analysis well.