Second-order Central Difference Approximation Research Articles

The discrete maximum principle and energy stability of a new second-order difference scheme for Allen-Cahn equations

This paper deals with the discrete maximum principle and energy stability of a new difference scheme for solving Allen-Cahn equations. By combining the second-order central difference approximation in space and the Crank-Nicolson method with Newton linearized technique in time, a two-level linearized difference scheme for Allen-Cahn equations is derived, which can yield accuracy of order two both in time and space. Under appropriate conditions, the scheme is proved to be uniquely solvable and able to preserve the maximum principle and energy stability of the equations in the discrete sense. With some numerical experiments, the theoretical results and computational effectiveness of the scheme are further illustrated.

Least-Square-Support-Vector-Machine-based approach to obtain displacement from measured acceleration

Recent advances in computer and sensing technologies have led to the proliferation of sensor networks in structural health monitoring and condition monitoring applications.Vibration data collected by sensors provide useful information about the condition of a structure or a machine component, facilitating identification of any changes in its performance. While acceleration and displacement data provide complementary information, a cost-effective alternative to monitoring both is to estimate displacements from accelerations. This paper presents a kernel regression approach for obtaining displacement time series from acceleration data. Starting from a second-order central difference approximation, the method performs ridge regression in a feature space induced by the linear kernel. The main advantages of the proposed method are (1) It does not require baseline adjustment, other than removing the mean of the acceleration record; (2) The solution obtained is numerically stable, and thus regularization is not necessary; (3) The reconstructed displacement does not exhibit any long period drift. The validity of the proposed method is demonstrated through examples, where structural systems’ displacements computed using the proposed approach were compared to the recorded experimental displacements. While the presented examples focus only on monitoring of vibrations responses of structural systems, the proposed method can be used in other settings where a displacement signal is to be estimated from an acceleration signal with appropriate, application-specific modifications.

Numerical simulation of 2-D laminar flow subjected to the Lorentz force effect in a channel with backward-facing step

This paper presents the numerical solutions of a two-dimensional laminar flow over a backward-facing step in the presence of the Lorentz body force. The Navier-Stokes equations in a vorticity-stream function formulation are numerically solved using a uniform grid mesh of 2001 × 51 points. A second-order central difference approximation is used for spatial derivatives. The solutions progress in time with a fourth-order Runge-Kutta method. The unsteady backward-facing step flow solution is computed for Reynolds numbers 100 to 800. The size and genesis of the recirculating regions are dramatically affected by applying the Lorentz force. The results demonstrate that using an appropriate configuration for applying the Lorentz force can make it an essential tool for controlling the flow in channels with a backward-facing step.

Numerical investigation of the Lorentz force effect on the vortex transfiguration in a two-dimensional lid-driven cavity

Numerical calculations of the two-dimensional unsteady incompressible driven cavity flow in the presence of the Lorentz body force are presented. The Navier—Stokes equations in vorticity-stream function formulation are solved numerically using a uniform grid mesh of 201×201. A second-order central difference approximation is used for spatial derivatives and the solutions march in time with a fourth-order Runge—Kutta method. The unsteady driven cavity flow solution is computed for a Reynolds number of Re=1000. The effects of the Stuart number (ratio of the electromagnetic forces to inertial forces) and penetration depth of the Lorentz force on the lid-driven cavity vortices are investigated. Two new quaternary vortices at the bottom corners of the cavity are observed in the flow field as the Stuart number and penetration depth increase. The primary vortex size also dwindles and eventually breaks down into two tertiary vortices as these parameters increase. The velocity component profiles are also considerably sensitive with respect to these parameters. It should be emphasized that on the basis of extensive comparisons and validations, which are made with benchmark solutions found in the literature, the numerical computations are repeated considering the Lorentz body force effect in the lid-driven cavity. Detailed results are presented in the article.

Computer simulation of turbulent flow through a hydraulic turbine draft tube

Based on the Navier-Stokes equations and the standardk-e turbulence model, this paper presents the derivation of the governing equations for the turbulent flow field in a draft tube. The mathematical model for the turbulent flow through a draft tube is set up when the boundary conditions, including the inlet boundary conditions, the outlet boundary conditions and the wall boundary conditions, have been implemented. The governing equations are formulated in a discrete form on a staggered grid system by the finite volume method. The second-order central difference approximation and hybrid scheme are used for discretization. The computation and analysis on internal flow through a draft tube have been carried out by using the simplec algorithm and cfx-tasc flow software so as to obtain the simulated flow fields. The calculation results at the design operating condition for the draft tube are presented in this paper. Thereby, an effective method for simulating the internal flow field in a draft tube has been explored.

Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates

A finite-difference method for solving three-dimensional time-dependent incompressible Navier–Stokes equations in arbitrary curvilinear orthogonal coordinates is presented. The method is oriented on turbulent flow simulations and consists of a second-order central difference approximation in space and a third-order semi-implicit Runge–Kutta scheme for time advancement. Spatial discretization retains some important properties of the Navier–Stokes equations, including energy conservation by the nonlinear and pressure-gradient terms. Numerical tests cover Cartesian, cylindrical-polar, spherical, cylindrical elliptic and cylindrical bipolar coordinate systems. Both laminar and turbulent flows are considered demonstrating reasonable accuracy and stability of the method.

A MAC SCHEME IN BOUNDARY-FITTED CURVILINEAR COORDINATES

Abstract An explicit finite-difference method has been developed for solving the equations of unsteady laminar natural convection in an arbitrarily shaped cavity. The method developed is an extension of the classical MAC (marker and cell) method to curvilinear quadralateral cells. These cells permit great geometric versatility. The boundary-fitted curvilinear coordinate system is used to generate a coordinate surface coincident with the boundary contours in the physical plane. The conservative form of the momentum equations in primitive variables are transformed to the rectangular computational plane. The MAC technique, formulated in the boundary-fitted coordinates, uses the contravariant velocity components to calculate the mass flux across the cells. A first-order forward difference approximation is used for the time derivative and second-order central difference approximation is used for the space derivatives. The finite-difference form of the continuity equation is satisfied for each cell, which is al...

An improved finite-difference approximation to the diffusion equation

A higher order approximation to field quantities than the piecewise constant approximation commonly used and implicit in the second-order central difference approximation to the Helmholtz equation is described. The continuous finite-difference (CFD) approximation results in reduced error without an increase in complexity of the associated computational algorithm.