An unfactored implicit scheme with multigrid acceleration for the solution of the Navier-Stokes equations

An unfactored implicit difference scheme for the steady state solution of the multidimensional Navier-Stokes equations of a compressible fluid is presented. The hyperbolic part is approximated by a high resolution scheme based on flux-vector splitting and upwind-biased differences to avoid the necessity of artificial dissipation terms and to construct a diagonal dominant solution matrix. Consequently, an iterative inversion of the solution matrix can be performed without any time step restriction. The rate of convergence is improved by using the indirect multigrid concept in form of the FAS scheme. The method is formulated for a body-fitted, curvilinear coordinate system. The computational results for laminar subsonic, transonic and supersonic steady-state flows which are compared with analytical and other numerical results as well as with experimental data illustrate the efficiency and the accuracy of the algorithm.     

An Efficient Correction Method to Obtain a Formally Third-Order Accurate Flow Solver for Node-Centered Unstructured Grids

A new method of obtaining third-order accuracy on unstructured grid flow solvers is presented. The method involves a simple correction to a traditional linear Galerkin scheme on tetrahedra and can be conveniently added to existing second-order accurate node-centered flow solvers. The correction involves gradients of the flux computed with a quadratic least squares approximation. However, once the gradients are computed, no second derivative information or high-order quadrature is necessary to achieve third-order accuracy. The scheme is analyzed both analytically using truncation error, and numerically using solution error for an exact solution to the Euler equations. Two demonstration cases for steady, inviscid flow reveal increased accuracy and excellent shock capturing with no loss in steady-state convergence rate. Computational timing results are presented which show the additional expense from the correction is modest compared to the increase in accuracy.     

Discrete Fundamental Solution Preconditioning for Hyperbolic Systems of PDE

We present a new preconditioner for the iterative solution of systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electromagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel that is a fundamental solution of a difference operator closely related to the original discretization. Analysis of a relevant scalar model problem in two spatial dimensions shows that grid independent convergence is obtained using a simple one-stage iterative method. As an example of a more involved problem, we consider the steady state solution of the non-linear Euler equations in a two-dimensional, non-axisymmetric duct. We present results from numerical experiments, verifying that the preconditioning technique again achieves grid independent convergence, both for an upwind discretization and for a centered second order discretization with fourth order artificial viscosity.     

Iterative methods for stationary solutions to the steady-state compressible Navier-Stokes equations

Numerical experiments are presented for the solution of the steady-state compressible Navier-Stokes equations. One test problem is fixed supersonic flow past a double ellipse, and the various solution methods studied. The problem is discretized using Osher's scheme, first- and second-order accurate. The fastest convergence to steady state is obtained using Newton's method. Simplifications of Newton's method based on domain decomposition are shown to perform well, whereas line relaxation methods meet with difficulties.     

Analysis of three stabilized finite volume iterative methods for the steady Navier–Stokes equations

In this work, three stabilized finite volume iterative schemes for the stationary Navier–Stokes equations are considered. Under the finite volume discretization at each iterative step, the iterative scheme I consists in solving the steady Stokes problem, iterative scheme II consists in solving the stationary linearized Navier–Stokes equations and iterative scheme III consists in solving the steady Oseen equations, respectively. We discuss the stabilities and convergence of three iterative methods. The iterative schemes I and II are stable and convergent under some strong uniqueness conditions, while iterative scheme III is unconditionally stable and convergent under the uniqueness condition. Finally, some numerical results are presented to verify the performance of these iterative schemes.     

A finite element first-order equation formulation for the small-disturbance transonic flow problem

The nonlinear, mixed elliptic hyperbolic equation describing a steady transonic flow is considered. The original equation is replaced by a system of first-order equations that are hyperbolic in time and defined in terms of velocity components. Parabolic regularization terms are added to capture shock wave solutions and to damp iterative solution algorithms. A finite element Galerkin method in space and a Crank-Nicolson finite difference method in iterative time are used to reduce the problem to the solution of a system of algebraic equations. Stability and convergence characteristics of the iterative method are discussed. The numerical implementation of the method is explained, and numerical results are presented.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

Numerical solution of laplace equation in three dimensions for a pierce type electron gun

In this paper, as part of a three-dimensional ribbon beam gun design, Laplace equation is solved in 3D for the case of boundaries which can be represented as simple surfaces which are subjected to specified voltages. Curve fitting by the least squares method is used to characterize the accelerator electrode shape as an extension of the 1D laminar electron flow in a Pierce type electron gun. The method involves solution of governing equations by an iterative finite difference technique. Modification of the standard Leibmann procedure is used to greatly increase the rapidity of convergence. The technique of handling irregular boundary points is considered in detail.     

A hybrid algorithm for approximate optimal control of nonlinear Fredholm integral equations

In this paper, a novel hybrid method based on two approaches, evolutionary algorithms and an iterative scheme, for obtaining the approximate solution of optimal control governed by nonlinear Fredholm integral equations is presented. By converting the problem to a discretized form, it is considered as a quasi-assignment problem and then an iterative method is applied to find an approximate solution for the discretized form of the integral equation. An analysis for convergence of the proposed iterative method and its implementation for numerical examples are also given.     

Finite-difference solution of the incompressible three-dimensional boundary layer equations for a blunt body

The governing equations for a laminar flow are solved in terms of an orthogonal surface coordinate system. One of the coordinate is determined by the intersection with the body surface of meridional planes which pass through an axis containing the stagnation point. The other coordinate is obtained numerically from the orthogonality condition. The momentum equations have been written in a standard from which allows additional equations of this form to be added with a small modification of the computer code. This equation is replaced with a nonlinear finite-difference equation which is solved as an iterative solution of linear tridiagonal equations. The special form of the governing equations at the stagnation point and the plane of symmetry is determined and the solution of these equations is obtained to provide a unified code. Numerical solutions have been obtained for several special cases and compared to results of other authors. New results are presented for an ellipsoid ar angle of attack and an elliptic-paraboloid at zero incidence.     

A multi-dimensional solver for the steady Euler equations

In this paper a numerical algorithm for the solution of the multi-dimensional steady Euler equations in conservative and non-conservative form is presented. Most existing standard and multi-dimensional schemes use flux balances with assumed constant distribution of variables along each cell edge, which interfaces two grid cells. This assumption is believed to be one of the main reasons for the limited advantage gained from multi-dimensional high order discretisations compared to standard one-dimensional ones. The present algorithm is based on the optimisation of polynomials describing the distribution of flow variables in grid cells, where only polynomials that satisfy the Euler equations in the entire grid cell can be selected. The global solution is achieved if all polynomials and by that the flow variables are continuous along edges interfacing neighbouring grid cells. A discrete approximation of a given spatial order is converged if the deviation between polynomial distributions of adjacent grid cells along the interfacing edge of the cells is minimal. Results from the present scheme between first and fifth order spatial accuracy are compared to standard first and second order Roe computations for simple test cases demonstrating the gain in accuracy for a number of sub- and supersonic flow problems.     

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v_θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.     

Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.     

The prediction of 2-dimensional recirculating flows using a simple finite-difference grid for non-rectangular flow fields

A finite-difference scheme for predicting 2-dimensional recirculating flows is extended by the use of a simple grid for non-rectangular flow fields. Examples of the grid layout and the appropriate finite-difference equations for laminar flow are presented. Also some of the important properties of the method are demonstrated by applying it to the prediction of a flow with an analytical solution.     

Numerical simulation of Newtonian and non-Newtonian flows in bypass

This paper deals with the numerical solution of Newtonian and non-Newtonian flows with biomedical applications. The flows are supposed to be laminar, viscous, incompressible and steady or unsteady with prescribed pressure variation at the outlet. The model used for non-Newtonian fluids is a variant of power law. Governing equations in this model are incompressible Navier–Stokes equations. For numerical solution we use artificial compressibility method with three stage Runge–Kutta method and finite volume method in cell centered formulation for discretization of space derivatives. The following cases of flows are solved: steady Newtonian and non-Newtonian flow through a bypass connected to main channel in 2D, steady Newtonian flow in angular bypass in 3D and unsteady non-Newtonian flow through bypass in 2D. Some 2D and 3D results that could have application in the area of biomedicine are presented.     

Accuracy and convergence of a finite element algorithm for laminar boundary layer flow

The Galerkin-weighted residuals formulation is employed to derive an implicit finite element solution algorithm for a generally non-linear initial-boundary value problem. Solution accuracy and convergence with discretization refinement are quantized in several error norms, for the non-linear parabolic partial differential equation system governing laminar boundary layer flow, using linear, quadratic and cubic functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equation appears extensible to the non-linear equations characteristic of laminar boundary layer flow.     

基于GaBP算法的快速潮流计算方法

研究在潮流迭代求解过程中雅可比矩阵方程组的迭代求解方法及其收敛性。首先利用PQ分解法进行潮流迭代求解,并针对求解过程中雅可比矩阵对称且对角占优的特性,对雅可比矩阵方程组采用高斯置信传播算法(GaBP)进行求解,再结合Steffensen加速迭代法以提高GaBP算法的收敛性。对IEEE118、IEEE300节点标准系统和两个波兰互联大规模电力系统进行仿真计算后结果表明：随着系统规模的增长,使用Steffensen加速迭代法进行加速的GaBP算法相对于基于不完全LU的预处理广义极小残余方法(GMRES)具有更好的收敛性,为大规模电力系统潮流计算的快速求解提供了一种新思路。     

Comparative study of pressure-correction and Godunov-type schemes on unsteady compressible cases

Two pressure-correction algorithms are studied and compared to an approximate Godunov scheme on unsteady compressible cases. The first pressure-correction algorithm sequentially solves the equations for momentum, mass and enthalpy, with sub-iterations which ensure conservativity. The algorithm also conserves the total enthalpy along a streamline, in a steady flow. The second pressure-correction algorithm sequentially solves the equations for mass, momentum and energy without sub-iteration. This scheme is conservative and ensures the discrete positivity of the density. Total enthalpy is conserved along a streamline, in a steady flow. It is numerically verified that both pressure-correction algorithms converge towards the exact solution of Riemann problems, including shock waves, rarefaction waves and contact discontinuities. To achieve this, conservativity is compulsory. The two pressure-correction algorithms and the approximate Godunov scheme are finally compared on cases with heat source terms: all schemes converge towards the same solution as the mesh is refined.     

Identification of the diagonal element to accelerate iterations in estimating the heat transfer in kinetic approximation

The paper examines modifications of an algorithm identifying the diagonal element for an iterative solution of implicit finite-difference equations approximating the nonlinear system of non-stationary differential equations consisting of the spectral integral-difference kinetic equation of the transfer of photons and the energy equation. Research is performed on the St scheme in plane geometry. Theoretical estimates of the convergence rate of the iterations and an example of the test problem are presented.     

The contrived transient-explicit method for solving steady-state flows: application to a rotating, recirculating flow

A method is described for solving steady-state fluid flow and heat transfer problems which are governed by elliptic-type differential equations. A contrived transient version of the steady-state problem is constructed by appending time derivatives to all the participating equations, regardless of whether or not such terms have physical reality. Each time derivative is multiplied by a fictive diffusivity coefficient which is varied during the course of an explicit marching procedure in order to achieve rapid, stable convergence to the steady state. The solution method is applied to a three-component laminar flow in a cylindrical enclosure having one rotating wall and coolant throughflow. Recirculation patterns are set up in the enclosure due to the shearing action of the throughflow and to the rotation of the disk. The surface heat transfer is found to decrease as the Reynolds number of the throughflow increases.