Piecewise Smooth Kernels Research Articles

An SPH stress correction algorithm based on the quartic piecewise smooth kernel function

<p indent="0mm">Smoothed particle hydrodynamics (SPH) is a popular adaptive Lagrangian mesh-free particle method. For the tensile instability problem in traditional SPH, according to the sufficient condition of tensile instability, a new quartic piecewise smooth kernel function with non-negative second derivative is proposed, and an SPH stress correction algorithm is obtained in this study. The numerical results show that the modified SPH algorithm can effectively describe the formation processes of 2D and 3D vdW liquid drops. Then, based on the modified SPH algorithm, the effects of different physical parameters on the particle distribution of 2D circular droplets are further discussed. The results show that, with the appropriate temperature or initial particle spacing, the boundary particles of circular droplets are neither clustered nor scattered, and as the fluid temperature or initial particle spacing increases, the boundary particles will be scattered. Conversely, the particle distribution of circular droplets becomes more uniform with the increase in the smooth length proportion coefficient.

Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library

This paper studies the second kind linear Volterra integral equations (IEs) with a discontinuous kernel obtained from the load leveling and energy system problems. For solving this problem, we propose the homotopy perturbation method (HPM). We then discuss the convergence theorem and the error analysis of the formulation to validate the accuracy of the obtained solutions. In this study, the Controle et Estimation Stochastique des Arrondis de Calculs method (CESTAC) and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library are used to control the rounding error estimation. We also take advantage of the discrete stochastic arithmetic (DSA) to find the optimal iteration, optimal error and optimal approximation of the HPM. The comparative graphs between exact and approximate solutions show the accuracy and efficiency of the method.

Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation

The evolutionary integral dynamical models of storage systems are addressed. Such models are based on systems of weakly regular nonlinear Volterra integral equations with piecewise smooth kernels. These equations can have non-unique solutions that depend on free parameters. The objective of this paper was two-fold. First, the iterative numerical method based on the modified Newton–Kantorovich iterative process is proposed for a solution of the nonlinear systems of such weakly regular Volterra equations. Second, the proposed numerical method was tested both on synthetic examples and real world problems related to the dynamic analysis of microgrids with energy storage systems.

Numerical approximation of first kind Volterra convolution integral equations with discontinuous kernels

The cubic ``convolution spline'' method for first kind Volterra convolution integral equations was introduced in P.J. Davies and D.B. Duncan, $\mathit {Convolution\ spline\ approximations\ of\ Volterra\ integral\ equations}$, Journal of Integral Equations and Applications \textbf {26} (2014), 369--410. Here, we analyze its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function and a new error bound obtained from a particular B-spline quasi-interpolant.

To the theory of volterra integral equations of the first kind with discontinuous kernels

A nonclassical Volterra linear integral equation of the first kind describing the dynamics of an developing system with allowance for its age structure is considered. The connection of this equation with the classical Volterra linear integral equation of the first kind with a piecewise-smooth kernel is studied. For solving such equations, the quadrature method is applied.

A modified Nyström–Clenshaw–Curtis quadrature for integral equations with piecewise smooth kernels

The Nyström–Clenshaw–Curtis (NCC) quadrature is a highly accurate quadrature which is suitable for integral equations with semi-smooth kernels. In this paper, we first introduce the NCC quadrature and point out that the NCC quadrature is not suitable for certain integral equation with well-behaved kernel functions such as e−|t−s|. We then modify the NCC quadrature to obtain a new quadrature which is suitable for integral equations with piecewise smooth kernel functions. Applications of the modified NCC quadrature to Wiener–Hopf equations and a nonlinear integral equation are presented.

On Some Classes of Linear Volterra Integral Equations

The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterra equation that appears in the integral models of developing systems. The Volterra integral equations of the first kind with piecewise smooth kernels are considered. Illustrative examples are presented.

A spectral collocation method for eigenvalue problems of compact integral operators

We propose and analyze a new spectral collocation method to solve eigenvalue problems of compact integral operators, particularly, piecewise smooth operator kernels and weakly singular operator kernels of the form 1 |t−s|μ , 0 < μ < 1. We prove that the convergence rate of eigenvalue approximation depends upon the smoothness of the corresponding eigenfunctions for piecewise smooth kernels. On the other hand, we can numerically obtain a higher rate of convergence for the above weakly singular kernel for some μ’s even if the eigenfunction is not smooth. Numerical experiments confirm our theoretical results.

On parametric families of solutions of Volterra integral equations of the first kind with piecewise smooth kernel

We suggest a method for constructing asymptotic approximations to parametric families of solutions of Volterra integral equations of the first kind with piecewise smooth kernel and prove the existence of continuous solutions.

A new formulation of the boundary integral equations of the first kind in electroelasticity

A system of boundary integral equations of the first kind with piecewise-smooth kernels, to which the boundary-value problems of electroelasticity reduce in the case of steady-state oscillations, is formulated. The proposed approach does not use the idea of fundamental solutions and is based solely on an anlysis of the characteristic polynomial of the electroelasticity operator