A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows

In this paper, we extend a previous work on a compact scheme for the steady Navier–Stokes equations [Li, Tang, and Fornberg (1995), Int. J. Numer. Methods Fluids, 20, 1137–1151] to the unsteady case. By exploiting the coupling relation between the streamfunction and vorticity equations, the Navier–Stokes equations are discretized in space within a 3×3 stencil such that a fourth order accuracy is achieved. The time derivatives are discretized in such a way as to maintain the compactness of the stencil. We explore several known time-stepping approaches including second-order BDF method, fourth-order BDF method and the Crank–Nicolson method. Numerical solutions are obtained for the driven cavity problem and are compared with solutions available in the literature. For large values of the Reynolds number, it is found that high-order time discretizations outperform the low-order ones.     

A Compact Higher Order Finite Difference Method for the Incompressible Navier–Stokes Equations

A numerical method based on compact fourth order finite difference approximations is used for the solution of the incompressible Navier–Stokes equations. Our method is implemented for two dimensional, curvilinear coordinates on orthogonal, staggered grids. Two numerical experiments confirm the theoretically expected order of accuracy.     

Navier-Stokes Solution by New Compact Scheme for Incompressible Flows

A new high spectral accuracy compact difference scheme is proposed here. This has been obtained by constrained optimization of error in spectral space for discretizing first derivative for problems with non-periodic boundary condition. This produces a scheme with the highest spectral accuracy among all known compact schemes, although this is formally only second-order accurate. Solution of Navier-Stokes equation for incompressible flows are reported here using this scheme to solve two fluid flow instability problems that are difficult to solve using explicit schemes. The first problem investigates the effect of wind-shear past bluff-body and the second problem involves predicting a vortex-induced instability.     

Numerical implementation of the asymptotic boundary conditions for steadily propagating 2D solitons of Boussinesq type equations

In the present paper, a difference scheme on a non-uniform grid is constructed for the stationary propagating localized waves of the 2D Boussinesq equation in an infinite region. Using an argument stemming form a perturbation expansion for small wave phase speeds, the asymptotic decay of the wave profile is identified as second-order algebraic. For algebraically decaying solution a new kind of nonlocal boundary condition is derived, which allows to rigorously project the asymptotic boundary condition at the boundary of a finite-size computational box. The difference approximation of this condition together with the bifurcation condition complete the algorithm. Numerous numerical validations are performed and it is shown that the results comply with the second-order estimate for the truncation error even at the boundary lines of the grid. Results are obtained for different values of the so-called ‘rotational inertia’ and for different subcritical phase speeds. It is found that the limits of existence of the 2D solution roughly correspond to the similar limits on the phase speed that ensure the existence of subcritical 1D stationary propagating waves of the Boussinesq equation.     

A Single-Step Characteristic-Curve Finite Element Scheme of Second Order in Time for the Incompressible Navier-Stokes Equations

In this paper we present a new single-step characteristic-curve finite element scheme of second order in time for the nonstationary incompressible Navier-Stokes equations. After supplying correction terms in the variational formulation, we prove that the scheme is of second order in time. The convergence rate of the scheme is numerically recognized by computational results.     

A high order finite-difference solver for investigation of disturbance development in turbulent boundary layers

This paper outlines a velocity–vorticity based numerical simulation method for modelling perturbation development in laminar and turbulent boundary layers at large Reynolds numbers. Particular attention is paid to the application of integral conditions for the vorticity. These provide constraints on the evolution of the vorticity that are fully equivalent to the usual no-slip conditions. The vorticity and velocity perturbation variables are divided into two distinct primary and secondary groups, allowing the number of governing equations and variables to be effectively halved. Compact finite differences are used to obtain a high-order spatial discretization of the equations. Some novel features of the discretization are highlighted: (i) the incorporation of the vorticity integral conditions and (ii) the related use of a co-ordinate transformation along the semi-infinite wall-normal direction. The viability of the numerical solution procedure is illustrated by a selection of test simulation results. We also indicate the intended application of the simulation code to parametric investigations of the effectiveness of spanwise-directed wall oscillations in inhibiting the growth of streaks within turbulent boundary layers.     

A Spectral Element Projection Scheme for Incompressible Flow with Application to the Unsteady Axisymmetric Stokes Problem

This paper presents a modified Goda scheme in the simulation of unsteady incompressible Navier–Stokes flows in cylindrical geometries. The study is restricted to the case of axisymmetric flows. For the justification of the robustness of our scheme some computational test cases are investigated. It turns out that by adopting the new approach, a significant accuracy improvement on both pressure and velocity can be obtained relative to the classical Goda scheme.     

A new fourth-order difference scheme for solving an N-carrier system with Neumann boundary conditions

In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods.     

A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted $L^2$- and $L^{\infty }$-norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order $min\{2?\alpha ,1+\alpha \}$, where $\alpha \in (0,1)$ is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.     

A Slowness Matching Eulerian Method for Multivalued Solutions of Eikonal Equations

Traveltime, or geodesic distance, is locally the solution of the eikonal equation of geometric optics. However traveltime between sufficiently distant points is generically multivalued. Finite difference eikonal solvers approximate only the viscosity solution, which is the smallest value of the (multivalued) traveltime (first arrival time). The slowness matching method stitches together local single-valued eikonal solutions, approximated by a finite difference eikonal solver, to approximate all values of the traveltime. In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation, so that the eikonal equation may be viewed as an evolution equation in one of the spatial directions. This paraxial assumption simplifies both the efficient computation of local traveltime fields and their combination into global multivalued traveltime fields via the slowness matching algorithm. The cost of slowness matching is on the same order as that of a finite difference solver used to compute the viscosity solution, when traveltimes from many point sources are required as is typical in seismic applications. Adaptive gridding near the source point and a formally third order scheme for the paraxial eikonal combine to give second order convergence of the traveltime branches.     

A parallel two-level method for simulating blood flows in branching arteries with the resistive boundary condition

Computer modeling of blood flows in the arteries is an important and very challenging problem. In order to understand, computationally, the sophisticated hemodynamics in the arteries, it is essential to couple the fluid flow and the elastic wall structure effectively and specify physiologically realistic boundary conditions. The computation is expensive and the parallel scalability of the solution algorithm is a key issue of the simulation. In this paper, we introduce and study a parallel two-level Newton–Krylov–Schwarz method for simulating blood flows in compliant branching arteries by using a fully coupled system of linear elasticity equation and incompressible Navier–Stokes equations with the resistive boundary condition. We first focus on the accuracy of the resistive boundary condition by comparing it with the standard pressure type boundary condition. We then show the parallel scalability results of the two-level approach obtained on a supercomputer with a large number of processors and on problems with millions of unknowns.     

A Comparative Study of Time Advancement Methods for Solving Navier–Stokes Equations

A qualitative and quantitative study is made for choosing time advancement strategies for solving time dependent equations accurately. A single step, low order Euler time integration method is compared with Adams–Bashforth, a second order accurate time integration strategy for the solution of one dimensional wave equation. With the help of the exact solution, it is shown that the presence of the computational mode in Adams–Bashforth scheme leads to erroneous results, if the solution contains high frequency components. This is tested for the solution of incompressible Navier–Stokes equation for uniform flow past a rapidly rotating circular cylinder. This flow suffers intermittent temporal instabilities implying presence of high frequencies. Such instabilities have been noted earlier in experiments and high accuracy computations for similar flow parameters. This test problem shows that second order Adams– Bashforth time integration is not suitable for DNS.