Fourth-order Accurate Finite Difference Method Research Articles

Simulating the near-field pulse-like ground motions of the Imperial Valley, California, earthquake

Long-period pulse-like ground motions in the near-field are likely to induce severe damage to large structures due to resonance at the natural frequencies of engineered construction. To investigate the origin and characteristics of large velocity pulses from the seismic source, this study identifies 14 stations with velocity pulses from 32 strong-motion seismographs located in the near-field of the Imperial Valley MW 6.5 earthquake. The pulse-like ground motions are simulated by a kinematic approach using a fourth-order accurate finite-difference method. The results show that the slip asperity in the northern part of the fault plane contributes more to large velocity pulses than the asperity that originates in the hypocentral zone. The two-sided pulses on the fault-normal component and the one-sided pulses on the fault-parallel component are mainly regulated by the fault rupture and slip directions, respectively. Compared with the pseudo-velocity response spectrum, the displacement response spectrum better captures the resonance effect of long-period pulse-like ground motion on large structures. The location of the maximum ground motion coincides with the surface projection area of the large asperity, and the amplitude and the attenuation rate of each horizontal component vary with increasing distance from the fault. The difference in the ground motion peaks on both sides of the fault implies that the fault-normal component is more sensitive to the fault dip angle than the fault-parallel component. We reproduce the pulse-like ground motions up to 1 Hz by fitting the simulated velocity time histories, response spectrum and ground motions to observed data. Using the pulse records of the Imperial Valley earthquake, the 1994 Northridge MW 6.7 earthquake, and the 1999 Chi-Chi MW 7.6 earthquake, we compare the effects of earthquake magnitude on peak value and distribution of large velocity pulses, in which a reasonable consistency can be observed. Our simulated results of near-field pulse-like ground motions are beneficial to clarify the pulse mechanism.

A fourth-order accurate finite difference method to evaluate the true transient behaviour of natural convection flow in enclosures

PurposeModelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.Design/methodology/approachFourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.FindingsError magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 103-108, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.Research limitations/implicationsThe developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.Originality/valueThe proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

Numerical Analysis of Three-Dimensional Sound Field in a Duct with Eccentric Slit-Like Expansion.

A numerical analysis is developed to the three-dimensional sound field in cylindrical duct with eccentric slit. A linearized unsteady Euler equation for acoustic disturbance is solved by a fourth-order accurate finite difference method. The boundary conditions at the curved rigid surface, such as the duct wall, are implemented properly for the linearized Euler equations. The three-dimensional structures of the sound fields are obtained for the several eccentricities of slit. The computed result shows that each eccentric slit has two different resonant frequencies, which are fully consistent with our previous experimental data.

A fourth-order accurate method for fourth-order two-point nonlinear boundary value problems

A fourth-order accurate finite difference method is developed for a class of fourth order nonlinear two-point boundary value problems. The method leads to a pentadiagonal scheme in the linear cases, which often arise in the beam deflection theory. The convergence of the method is tested numerically on examples from the literature.