Higher-order Accurate Algorithms Research Articles

An Improved Higher-Order Time Integration Algorithm for Structural Dynamics

Based on the weighted residual method, a single-step time integration algorithm with higher-order accuracy and unconditional stability has been proposed, which is superior to the second-order accurate algorithms in tracking long-term dynamics. For improving such a higher-order accurate algorithm, this paper proposes a two sub-step higher-order algorithm with unconditional stability and controllable dissipation. In the proposed algorithm, a time step interval [tk, tk + h] where h stands for the size of a time step is divided into two sub-steps [tk, tk + γh] and [tk + γh, tk + h]. A non-dissipative fourth-order algorithm is used in the rst sub-step to ensure low-frequency accuracy and a dissipative third-order algorithm is employed in the second sub-step to lter out the contribution of high-frequency modes. Besides, two approaches are used to design the algorithm parameter γ. The rst approach determines γ by maximizing low-frequency accuracy and the other determines γ for quickly damping out highfrequency modes. The present algorithm uses ρ∞ to exactly control the degree of numerical dissipation, and it is third-order accurate when 0 ≤ ρ∞ < 1 and fourth-order accurate when ρ∞ = 1. Furthermore, the proposed algorithm is self-starting and easy to implement. Some illustrative linear and nonlinear examples are solved to check the performances of the proposed two sub-step higher-order algorithm.

A higher-order accurate algorithm based on two-dimensional unstructured mesh for simulation of charge transportation in solid dielectrics

In a high voltage direct current (HVDC) transmission system, the devices like HVDC cable and its accessory, HVDC bush play important roles in power transmission and power transformation. The local electric field could be greatly enhanced due to the space charge accumulation in the insulation, which significantly influences the electrical performance of the electric devices. In this paper, an algorithm for solving the charge coupled systems of solid dielectrics is presented. The algorithm is based upon a finite element method (FEM) solving the Poisson equation and a finite volume method (FVM) discretizing the transportation equation on a cell-centered tessellation. Both FEM and FVM share a set of unstructured mesh. The study focuses on the capability of different upwind schemes on capture of charge density step. The numerical experiment results show that in contrast to conventional methods that constrain charge densities to piecewise constants within a cell, a higher-order approximation of charge density is obtained by reconstructing the charge densities gradient in the control volumes. And the oscillation could be avoided when a limiter function is adopted to make sure that no new extrema created between the central cell and the neighbor cells. The influences of time step, total calculation time, and the mesh structured or not are also investigated. The results show that the algorithm could provide enough accuracy when suitable time step and time duration are adopted with unstructured mesh.

SHARP: A Spatially Higher-order, Relativistic Particle-in-cell Code

Abstract Numerical heating in particle-in-cell (PIC) codes currently precludes the accurate simulation of cold, relativistic plasma over long periods, severely limiting their applications in astrophysical environments. We present a spatially higher-order accurate relativistic PIC algorithm in one spatial dimension, which conserves charge and momentum exactly. We utilize the smoothness implied by the usage of higher-order interpolation functions to achieve a spatially higher-order accurate algorithm (up to the fifth order). We validate our algorithm against several test problems—thermal stability of stationary plasma, stability of linear plasma waves, and two-stream instability in the relativistic and non-relativistic regimes. Comparing our simulations to exact solutions of the dispersion relations, we demonstrate that SHARP can quantitatively reproduce important kinetic features of the linear regime. Our simulations have a superior ability to control energy non-conservation and avoid numerical heating in comparison to common second-order schemes. We provide a natural definition for convergence of a general PIC algorithm: the complement of physical modes captured by the simulation, i.e., those that lie above the Poisson noise, must grow commensurately with the resolution. This implies that it is necessary to simultaneously increase the number of particles per cell and decrease the cell size. We demonstrate that traditional ways for testing for convergence fail, leading to plateauing of the energy error. This new PIC code enables us to faithfully study the long-term evolution of plasma problems that require absolute control of the energy and momentum conservation.

IDeC( k): A new velocity reconstruction algorithm on arbitrarily polygonal staggered meshes

We propose a novel algorithm for velocity reconstruction from staggered data on arbitrary polygonal staggered meshes. The formulation of the new algorithm is based on a constant polynomial reconstruction approach in conjunction with an iterative defect correction method and is referred to as the IDeC( k) reconstruction. The algorithm is designed for second order accuracy of the reconstructed velocity field and also leads to a consistent estimate of velocity gradients. Accuracy, convergence and robustness of the new algorithm are studied on different mesh topologies and the need for higher-order reconstruction is demonstrated. Numerical experiments for several cases including incompressible viscous flows establish the IDeC( k) reconstruction as a generic, fast, robust and higher-order accurate algorithm on arbitrary polygonal meshes.

Unconditionally stable higher-order accurate collocation time-step integration algorithms for first-order equations

In this paper, unconditionally stable higher-order accurate time-step integration algorithms for linear first-order differential equations based on the collocation method are presented. The amplification factor at the end of the spectrum is a controllable algorithmic parameter. The collocation parameters for unconditionally stable higher-order accurate algorithms are found to be given by the roots of a polynomial in terms of the ultimate amplification factor. In general, when the numerical solution is approximated by a polynomial of degree n, this approximation is at least nth order accurate. However, by using the above collocation parameters, the order of accuracy can be improved to 2 n−1 or 2 n. The approximate solutions are found to be equivalent to the generalized Padé approximations. Furthermore, it is shown that the accuracy of the particular solution due to excitation given by the present method is compatible with the homogeneous solutions. No modification of the collocation parameters is required.

MIXED TIME FINITE ELEMENTS FOR VIBRATION RESPONSE ANALYSIS

This paper presents a mixed time finite element method for vibration response analysis. The underlying variational formulation is developed from the principle of virtual work. Conventional spatial discretization techniques are adopted. Time discretization is implemented by using mixed time finite elements. The proposed formulation not only forms a unified variational basis for spatial and temporal discretization; it also derives many other robust time step integration algorithms. Unconditionally stable and high order accurate algorithms with variable numerical dissipation can be constructed systematically. Moreover, the proposed formulations can be used to solve linear as well as non-linear problems. The accuracy and stability of the derived algorithms are presented.