High-accuracy Finite Difference Method Research Articles

Effects of inclination angle and fluid parameters on binary fluid convection in a tilted rectangular cavity

Thermal convection of binary fluid mixtures with negative separation ratios in a tilted rectangular cavity heated from below is numerically investigated using a high-accuracy finite-difference method. The fluids under consideration are Newtonian fluids with the Prandtl number (Pr) and Lewis number (Le) in the ranges of 0.01≤Pr≤10 and 0.001≤Le≤1, respectively. Extensive direct numerical simulations are conducted to explore the dynamics of the system by varying the control parameters, namely, the relative Rayleigh number (r) and the inclination angle of the cavity (β) together with the separation ratio (ψ) in the ranges of 1.0≤r≤2.0, 0°≤β≤90°, and −0.6≤ψ≤0. It is found that the inclination angle leads to a distinct trend for heat and mass transfer: the Nusselt number first decreases for small β, then gradually increases for moderate β, and finally decreases again for large β when r varies from 1.4 to 2.0. The optimal inclination angle lies in the range of 54°≤β∗≤64°, where the heat transport peaks. The Sherwood number initially decreases for small β, then increases with increasing β, and finally remains constant for large β. We present the bifurcation diagram of the solutions for different separation ratios, focusing on the distinct variation trends of the inclination angle. The corresponding heat- and mass-transfer properties of the flow on different branches of the bifurcation diagram are investigated. Furthermore, we explore the effect of the relative Rayleigh number, Prandtl number and Lewis number as well as the separation ratio on heat and mass transfer for various inclination angles, and obtain the Nusselt number as a function of the separation ratio and inclination angle.

Two-layer Boussinesq models for coastal water waves

This paper presents three sets of two-layer Boussinesq models for highly dispersive and highly nonlinear water waves. These models are formulated in terms of depth-averaged velocities or velocities located at two arbitrary z locations within each layer and are fully nonlinear to the second order. Stokes-type expansions are used to theoretically analyze the linear and nonlinear properties of the models. The coefficients involved in the governing equations are determined from the minimization of the integral error between the linear wave celerity, shoaling gradient, second nonlinear harmonics of the equations and the related analytical solutions. The most promising model is applicable up to kh≈25.4 (where kh is the dimensionless water depth, k is the wave number, and h is the water depth) for the dispersive property, to kh≈6 for the second nonlinear property within 1% error, and to kh≤6 for the excellent shoaling property. The numerical implementation for one-dimensional governing equations on non-staggered grids is also presented by employing a fourth-order Adams–Bashforth–Moulton time integration and the high-accuracy finite difference method. Four demanding numerical experiments that require high accuracy of dispersion and nonlinearity are conducted to assess the performance of the models. The computational results are compared against the analytical solution and experimental data, good agreements are found.

A High Accuracy Finite Difference Method for Linear Fifth Order Boundary Value Prolbems

In this paper, some high accuracy finite difference schemes for solving the fifth order linear boundary value problem and the derivatives of the solutions are presented. Some appropriate examples are employed to evaluate the performance of the proposed element. The numerical results show that finite difference method can obtain high accuracy solutions.

Onset of cellular flame instability in adiabatic CH4/O2/CO2 and CH4/air laminar premixed flames stabilized on a flat-flame burner

This paper presents numerical and experimental studies on the onset and evolution of laminar cellular flames of CH4/O2/CO2 (oxy-fuel) and CH4/air mixtures under adiabatic conditions, stabilized in the proximity of a flat-flame burner at atmospheric pressure. In the numerical simulations, a two-dimensional domain with periodicity at the transverse far field boundaries is resolved using a high accuracy finite difference method and employing a detailed chemical kinetic mechanism and detailed transport properties. In the experiments a specially designed adiabatic flat-flame burner, a so-called heat flux burner, is employed. A key parameter, the standoff distance between the flame front and the burner exit plate, is identified. A critical standoff distance is found, above which cellular flame instability is observed. It is shown that the critical standoff distance is closely related to the density ratio and the laminar flame thickness for each flame studied. The observed onset of cellular flames is governed by the hydrodynamic instability mechanism, which is generally suppressed by the burner when the flame is very close to the burner plate. Diffusive-thermal effects play an important role in the flame instability when the flame is far from the burner. The critical standoff distance has no clear correlation with the Lewis number, indicating a less significant effect of diffusive-thermal instability on the flames near the burner.

High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form

In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u´)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.

Analysis of anisotropy of numerical wave solutions by high accuracy finite difference methods

Fourier analysis is used to quantitatively assess the resolution, and in particular the isotropy of wave solution using finite difference spatial discretization schemes along with fourth order Runge–Kutta temporal scheme. Aspect ratio of the grid in two-dimension, along with the angle of wave propagation are the parameters varied to qualitatively and quantitatively assess the anisotropy of the solutions for (a) a skewed one-dimensional wave convecting in two-dimensions following the standard convection equation and (b) a wave propagating following the two-dimensional linearized rotating shallow water equations. Results show the effect of changing the aspect ratio and the propagation angle on the directional nature of the solution as obtained by different methods for the above non-dispersive and dispersive wave system.

High-accuracy finite-difference methods for the valuation of options

A finite-difference scheme often employed for the valuation of options from the Black–Scholes equation is the Crank–Nicolson (CN) scheme. The CN scheme is second order in both time and asset. For a rapid valuation with a reasonable resolution of the option price curve, it requires extremely small steps in both time and asset. In this paper, we present high-accuracy finite-difference methods for the Black–Scholes equation in which we employ the fourth-order L-stable Simpson-type (LSIMP) time integration schemes developed earlier and the well-known Numerov method for discretization in the asset direction. The resulting schemes, called LSIMP–NUM, are fourth order in both time and asset. The LSIMP–NUM schemes obtained can provide a rapid, stable and accurate resolution of option prices, allowing for relatively large steps in both time and asset. We compare the computational efficiency of the LSIMP–NUM schemes with the CN and Douglas schemes by considering valuation of European options and American options via the linear complementarity approach.

Numerical Solution of the Time-Domain Maxwell Equations Using High-Accuracy Finite-Difference Methods

High-accuracy finite-difference schemes are used to solve the two-dimensional time-domain Maxwell equations for electromagnetic wave propagation and scattering. The high-accuracy schemes consist of a seven-point spatial operator coupled with a six-stage Runge--Kutta time-marching method. Two methods are studied, one of which produces the maximum order of accuracy and one of which is optimized for propagation distances smaller than roughly 300 wavelengths. Boundary conditions are presented which preserve the accuracy of these schemes when modeling interfaces between different materials. Numerical experiments are performed which demonstrate the utility of the high-accuracy schemes in modeling waves incident on dielectric and perfect-conducting scatterers using Cartesian and curvilinear grids. The high-accuracy schemes are shown to be substantially more efficient, in both computing time and memory, than a second-order and a fourth-order method. The optimized scheme can lead to a reduction in error relative to the maximum-order scheme, with no additional expense, especially when the number of wavelengths of travel is large.

High-accuracy bench mark solutions to natural convection in a square cavity

A fourth-order high-accuracy finite difference method is presented for the bouyancy-driven flow in a square cavity with differentially heated vertical walls. The two bench mark solutions against which other solutions can be compared were obtained. The present solution is seemed to be accurate up to fifth decimal. The proposed scheme is stable and convergent for high Rayleigh number, and will be applicable to general problems involving flow and heat transfer, especially in three dimensions.

Highly accurate compact implicit methods and boundary conditions

Compact methods are high accuracy finite-difference methods where the functions and their derivatives are considered as unknowns. Two methods are presented to eliminate the second-order derivatives in parabolic equations, while keeping the fourth-order accuracy and the tridiagonal nature of the schemes. A type of high accuracy additional boundary condition is also proposed, which is consistent with the high accuracy of the inner scheme and uses only values at interior and boundary nodes. Integration on nonregular meshes is also examined.

On a high accuracy finite difference method

A highly accurate finite difference method is proposed for the numerical solution of partial differential equations that describe initial value problems. It includes a spatial smoothing operation and avoids the computational mode. The method is presented here for one space dimension and is tested on a linear equation, where it gives highly accurate results for the computed phase speed. The solution of a nonlinear equation confirms unconditional nonlinear stability.