Finite Difference Relaxation Research Articles

Calculation of the ion current to a conducting cylinder in a supersonic flow of a collisionless plasma

The diagnostics of low-temperature plasma flows using cylindrical probes is based on the classical Langmuir relation for the ion current to a thin, in comparison with the Debye length, cylinder. The aim of this work is to study the applicability of the Langmuir relation for a cylinder whose radius exceeds the Debye length. The interaction of a conducting cylinder with a rarefied plasma flow was simulated numerically. The cylinder had a negative potential with respect to the plasma. Free molecular flow around the cylinder was simulated on the basis of a two-dimensional system of the Vlasov–Poisson equations. The electron-repulsing local equilibrium self-consistent electric field was calculated using the Poisson–Boltzmann model in the approximation of local equilibrium electrons and taking into account an electron sink on the cylinder surface in the central field approximation. The Vlasov equations for ions and the Poisson–Boltzmann equations for the self-consistent electric field were solved on nested grids by a finite-difference relaxation method with splitting by physical processes and using the method of characteristics. The reliability of the calculated results was confirmed by the solution of known model problems and a comparison with the results of other authors and the results of solving identical physical problems with the use of different mathematical models and methods. The ion current to a cylinder placed transversely to a plasma flow was calculated as a function of the cylinder potential, the ion velocity ratio, and the ratio of the characteristic dimension of the cylinder to the Debye length. From the calculated results, numerical estimates were obtained for the range of applicability of the classical Langmuir relation for the ion current to a cylinder whose radius exceeds the Debye length. The results obtained may be used in the diagnostics of supersonic flows of a low-temperature rarefied plasma.

High order approach for solving chaotic and hyperchaotic problems

In this work, the method of Taylor's decomposition on two points is suggested in order to find approximate solutions of chaotic and hyperchaotic initial value problems and to analyze the behaviors of these solutions. Unlike to the classical Taylor's method, the proposed numerical scheme is based on the application of the Taylor's decomposition on two points to the system of nonlinear initial value problems, and as a result an implicit method is obtained. Stability and error analysis of the method are presented, and its high-order accuracy and A-stability are proven. One of the advantages of the proposed method is that it is a stable and very efficient method for chaotic problems as it is an implicit one-step method. The most important advantage of the Taylor's decomposition method is that it has high order accuracy for large step sizes with a simple algorithm compared to other methods. The applicability of the proposed method has been examined in some famous chaotic systems; the Lorenz and Chen systems, and hyperchaotic systems; the Chua and Rabinovich-Fabrikant systems, to emphasize both its accuracy and effectiveness. The accuracy of the proposed method is checked by comparing the calculated results with semi-explicit Adams-Bashforth-Moulton method and ninth order Runge-Kutta method. The calculated results are also compared with multi-stage spectral relaxation method and multi-domain compact finite difference relaxation method. Comparisons have shown that the method is more accurate and efficient than the other mentioned methods for large step sizes. The obtained results are also compared with the theoretical findings and it is shown that the theoretical and numerical results are consistent.

Numerical Solutions of Nine-Dimensional Lorenz System

In this study, the performance of the Taylor’s decomposition method in the nine-dimensional Lorenz system is investigated. The proposed method is applied to the nine-dimensional Lorenz system for both chaotic and hyperchaotic cases. Phase portraits and time series graphs of the problem are drawn. The results obtained are compared with both the fourth order Runge-Kutta method and multi-domain compact finite difference relaxation method. Comparisons show that the obtained results are consistent with the results of other methods. Thus, the accuracy and efficiency of the method for nonlinear systems were emphasized by using the nine-dimensional Lorenz system.

On the multi-domain compact finite difference relaxation method for high dimensional chaos: The nine-dimensional Lorenz system

In this paper, we implement the multidomain compact finite difference method to numerically study high dimensional chaos by considering the nine-dimensional Lorenz system. Most of the existing numerical methods converge slowly for this kind of problems and this results in inaccurate approximations. Though highly accurate, the compact finite difference method becomes less accurate for problems characterized by chaotic solutions, even with an increase in the number of grid points. As a result, in this work, we adopt the multidomain approach. This approach remarkably improves the results as well as the efficiency of the method.

Compact finite difference relaxation method for chaotic and hyperchaotic initial value systems

In this paper, we present a new application of higher order compact finite differences to solve nonlinear initial value problems exhibiting chaotic behaviour. The method involves dividing the domain of the problem into multiple sub-domains, with each sub-domain integrated using higher order compact finite difference schemes. The nonlinearity is dealt using a Gauss–Seidel-like relaxation. The method is, therefore, referred to as the multi-domain compact finite difference relaxation method (MD-CFDRM). In this new application, the MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. The main advantage of the new approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method. The results are compared with spectral-based multi-domain method.

Finite-Difference Relaxation for Parallel Computation of Ionized Field of HVDC Lines

Ionized field calculations for high-voltage direct current (HVDC) transmission line is a computationally demanding problem, which can benefit from the application of massively parallel high-performance compute architectures. The finite element method (FEM) commonly employed to solve this problem is both memory and execution time intensive. In this paper, a finite-difference relaxation (FDR) method is proposed to solve a unipolar and a bipolar ionized field problem in an HVDC line. The novel FDR method has several advantages over FEM. First, the scheme is suitable for massively parallel computation and runs much faster: Compared with the commercial FEM software Comsol Multiphysics, the speed-up is more than 14 times in CPU parallelization and 35 times in graphics processor parallel implementation, while providing high accuracy. Moreover, the set of equations in FDR need not be assembled; instead, it is solved by a relaxation scheme and requires much less memory than FEM. Additionally, differentiated grid size with interpolation techniques is proposed to improve the flexibility of FDR for problem domain containing irregular geometries or disproportional sizes.

Finite-Difference Relaxation for Parallel Computation of Ionized Field of HVDC Lines

Finite-Difference Relaxation for Parallel Computation of Ionized Field of HVDC Lines

Two-dimensional flows in finite-width channels partially filled with porous medium

A stationary two-dimensional liquid flow in a channel partially filled with a porous non-deformable homogeneous medium is investigated numerically based on the proposed mathematical model. The homogeneous liquid is bounded by a solid or free rigid boundary, while the channel bottom is solid. The model is constructed based on the Berman transformation. The free flow is described by the Navier-Stokes equations and the filtration flow is described by the Darcy-Brinkman model. The interfacial boundary conditions include the velocity continuity and the balance of tangential and normal viscous stresses; the tangential stresses may have a discontinuity at the boundary. The numerical solution is found by the finite-difference relaxation method. It has been shown that the flow of the liquid into porous medium takes place for a wide parameter range. However in most cases, the transversal velocity is about 10-7of the maximum longitudinal velocity. The dependence of the transversal velocity on the permeability of porous medium and its thickness is examined. The velocity profiles are plotted for different sets of parameters. The effect of the relative thickness of porous medium on the transversal velocity maximum and its position is also considered. It has been found that the effect of the transversal flow becomes significant if the width of the free portion of the channel is less than 10% of the width of the entire system. In spite of the smallness of the transversal velocity, the total flow rate through the interface is about 0.1% of the total flow rate through the channel provided that the channel length is 105÷106of the channel width.

Pipelining granularity optimization algorithm based on loop tiling

When the pipelining loop has a great number of iterations,or the size of its body is large,but the number of available threads is small,the workload between two synchronizations of a thread is so heavy,which produces pretty low degree of parallelism.The traditional trade-off approach based on loop tiling cannot handle the above situation.To solve this problem,a pipelining granularity decreasing approach based on loop tiling was proposed.The optimal pipelining granularity was obtained by building the cost model for pipelining loop and a pipelining granularity optimizing algorithm was implemented.By measuring the wavefront loops of Finite Difference Relaxation(FDR) and the representative loops of Finite Difference Time Domain(FDTD),the loops show better performance improvement by using the proposed algorithm than the traditional one.

On the Comparison between Compact Finite Difference and Pseudospectral Approaches for Solving Similarity Boundary Layer Problems

We introduce two methods based on higher order compact finite differences for solving boundary layer problems. The methods called compact finite difference relaxation method (CFD-RM) and compact finite difference quasilinearization method (CFD-QLM) are an alternative form of the spectral relaxation method (SRM) and spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral-based methods which have been successfully used to solve boundary layer problems. The main objective of this paper is to give a comparison of the compact finite difference approach against the pseudo-spectral approach in solving similarity boundary layer problems. In particular, we seek to identify the most accurate and computationally efficient method for solving systems of boundary layer equations in fluid mechanics. The results of the two approaches are comparable in terms of accuracy for small systems of equations. For larger systems of equations, the proposed compact finite difference approaches are more accurate than the spectral-method-based approaches.

Higher Order Compact Finite Difference Schemes for Unsteady Boundary Layer Flow Problems

We investigate the applicability of the compact finite difference relaxation method (CFDRM) in solving unsteady boundary layer flow problems modelled by nonlinear partial differential equations. The CFDRM utilizes the Gauss-Seidel approach of decoupling algebraic equations to linearize the governing equations and solve the resulting system of ordinary differential equations using compact finite difference schemes. The CFDRM has only been used to solve ordinary differential equations modelling boundary layer problems. This work extends its applications to nonlinear partial differential equations modelling unsteady boundary layer flows. The CFDRM is validated on two examples and the results are compared to results of the Keller-box method.

Regularized additive operator-difference schemes

The construction of additive operator-difference (splitting) schemes for the approximate solution Cauchy problem for the first-order evolutionary equation is considered. Unconditionally stable additive schemes are constructed on the basis of the Samarskii regularization principle for operator-difference schemes. In the case of arbitrary multicomponent splitting, these schemes belong to the class of additive full approximation schemes. Regularized additive operator-difference schemes for evolutionary problems are constructed without the assumption that the regularizing operator and the operator of the problem are commutable. Regularized additive schemes with double multiplicative perturbation of the additive terms of the problem’s operator are proposed. The possibility of using factorized multicomponent splitting schemes, which can be used for the approximate solution of steadystate problems (finite difference relaxation schemes) are discussed. Some possibilities of extending the proposed regularized additive schemes to other problems are considered.

Landau–de Gennes modelling of nematic liquid crystal colloids

Phenomenological Landau–de Gennes modelling based on the free energy of nematic liquid crystal colloids is reviewed. Nematic phase, gradient of order, and surface anchoring contributions to the total free energy are used. The numerical finite difference relaxation technique is explained as an efficient tool for the minimisation of the free energy. Effects of the mesh and mesh allocation are discussed. Various conceptually different colloidal structures are calculated to show the universality of the model. Single particles, dipolar–quadrupolar dimers, entangled dimers, dimers bound by escaped hyperbolic rings, and hierarchically patterned Saturn-ring colloidal superstructures are presented.

Dynamics of spin-dependent tunneling through a semiconductor double-barrier structure

The dynamics of spin-dependent tunneling through a nonmagnetic semiconductor double-barrier structure is studied including the k3 Dresselhaus spin orbit coupling is solved by the time-dependent Schrödinger equation with a developed method for the finite-difference relaxation. The resonant peak and quasibound level lifetime are determined by the in-plane wave vector and the applied electric field. The buildup time and decay lifetime of resonant probability amplitude are different for the spin-down and spin-up electrons due to the Dresselhaus spin-orbit coupling. Further investigation shows that the steady spin-polarization in both the well and collector regions has been obtained in the time domain.

Concurrent processing applied to the Exodus Monte Carlo technique

AbstractThis paper illustrates how problems based on the Probabilistic Potential Method can be solved using a parallelized form of the Exodus Monte Carlo technique. The Exodus technique which does not use random number generation is shown to be more efficient for moderately large problems than the conventional fixed random walk method when programmed in both sequential and concurrent forms.The technique is illustrated for an electrostatic field problem by using as an example internal potential evaluation for a rectangular coaxial line. The problem geometry is torn into subsections which are then analysed independently using concurrent programming methods. The resulting algorithm is compared with the same problem executed using a conventional sequential algorithm. Computation is achieved by use of an array of transputers. The design of the concurrent algorithm is described in detail and typical results are compared with the fixed random walk Monte Carlo method and also with the finite difference relaxation method. Parallel Fortran is used throughout as the programming language.

Liquid metal flow through a sharp elbow in the plane of a strong magnetic field

This paper treats a liquid-metal flow through a sharp elbow connecting two constant-area, rectangular ducts with thin metal walls. There is a uniform, strong magnetic field in the plane of the centrelines of the ducts. An analytical solution with a series of eigenfunctions is possible for two sectors of the geometry, while a finite-difference relaxation solution is used for the third sector. The analytical and numerical solutions are coupled at the common boundaries by a combination of a Galerkin minimization of a residual and of integrals of the basic conservation laws over cells adjacent to each boundary. Results are presented for the three-dimensional pressure, electric potential function and fluid velocity. The pressure drop due to the three-dimensional effects near the elbow is also presented. The eigenfunction series represents a quite general solution for any three-dimensional flow in a rectangular duct with a skewed magnetic field.

Some questions concerning the initial fields in finite difference computation of two-dimensional steady transonic flows

Abstract To study the influence of initial fields on computation results, the author, starting from a two-dimensional steady transonic small-disturbance equation, has computed the shock wave strengths and lift coefficients for the NACA0012 airfoil by the mixed finite difference relaxation iteration method. The computed results show that if (1) incident flows are supercritical, (2) a nonconservative difference scheme is used to capture shock waves, and (3) the computations are iterated with the initial fields at the lower entropy values (zero values or computed results of flow fields with smaller angles of attack or Mach numbers), then correct converged solutions can be obtained. Otherwise, erroneous converged solutions might be obtained. If incident flows are subcritical and shock waves do not appear in the flow field, then the computed results do not depend on the choice of the initial fields. This means that the converged solution is unique.

Axisymmetric transonic flow past slender bodies, including perforated wall interference effects

Solutions of the transonic small-perturbation equation for flow past slender bodies of revolution at subsonic freestream Mach numbers are presented in free air as well as in the presence of perforated walls. An artificial viscosity term which enables shock capture is added explicitly to the small-perturbation equation. The modified equation is then converted into an integral equation by the use of Green's theorem. Numerical solution to the integral equation is obtained by discretizing the region of integration into rectangular panels wherein the flow quantities can be considered uniform. Type-dependent operators are introduced in the calculation of the nonlinear source term. The resulting system of algebraic equations is then iteratively solved either by a single- step direct iteration scheme or a quasi-Newton scheme. HE finite difference relaxation procedure has been ap- plied to axisymmetric flow around bodies of revolution by Bailey 1 and Krupp and Mmrman2 and compared with the experimental data of McDevitt and Taylor.3'4 Using the method of split potential Sedin5 has devised a stable iterative scheme for the same problem. Both Bailey and Sedin have included tunnel wall interference effects. They have used the classical homogeneous wall boundary condition developed by Baldwin et al. 6 to model the presence of a perforated wall. Bailey's theoretical calculations seem to confirm the validity of this formulation insofar as the theoretical pressure distribution shows good agreement with the measured values and the shock location is correctly predicted for the ex- perimentally determined value of the porosity parameter. Thus, the theoretical model on which the computation of axisymmetric transonic flow with tunnel wall boundaries is based appears to be reasonably effective in predicting the flow details. It is well known that the axisymmetric transonic small- perturbation equation (TSPE), together with the linearized boundary condition on the body surface and a vanishing far- field disturbance, can be converted into an integral equation by the application of Green's theorem to the flow domain exterior to the body.7 In this formulation, the body boundary condition appears as a line integral on the body contour. Because the line integral for the axisymmetric flow is in- dependent of the unknown nonlinear field source term, iterative readjustment of the boundary condition is not necessary. Since the accuracy with which the tangency con- dition is satisfied depends only on the accuracy of the evaluation of the line integral, choice of the grid details in the vicinity of the body is unimportant to the overall com- putation. Further, since the far-field conditions are implicitly satisfied by the integral equation, no matching is necessary at the far-field boundary as is generally the case in finite difference calculations. The grid boundary is chosen after carrying out numerical experiments to assess its influence on the surface pressure distribution; the number of grid points can thus be minimized. The presence of the wall gives rise to an additional line integral on the wall, with appropriate changes in the domain of integration for the field integral.

Transonic Flutter Analysis of a Rectangular Wing with Conventional Airfoil Sections

Flutter analysts have encountered considerable analytical difficulties in the prediction of the flutter stability of aircraft operating in the transonic Mach number regime. Because of the shocks and nonlinearities of transonic flow the aerodynamic unsteady forces have been difficult to determine and have prohibited accurate determination of the flutter speed. The finite-difference relaxation method is used to determine the oscillatory transonic aerodynamic forces on a uniformly stiff cantilever rectangular wing in a flow field with mixed subsonic and supersonic regions together with shock waves. The flutter speed is determined at two transonic Mach numbers and is compared to the flutter speed obtained using a linear aerodynamic theory.

Recent developments in integral equation method in transonic flow

The present work gives a brief review of the integral equation method in transonic aerodynamics, with particular emphasis on the works of Nϕrstrud, Nixon and Niyogi. The use of integral equation method can result in significant reduction in computer time, roughly by a factor of 60, compared to the finite difference relaxation procedure. In shock-free symmetrical super-critical flows, the simple solution of Niyogi which delivers results with less than 5 % overall error, may be conveniently used for analysis as well as for design purposes. The direct iteration scheme, applicable to shock-free flow, as well as to flows with shocks, emerges as particularly attractive due to its favourable convergence behaviour and computational simplicity. It is expected to be even more effective for three-dimensional problems. Further, the simple model suggested by Oswatitsch for thin symmetrical profile flow is sufficiently accurate for most practical purposes. For results of high accuracy, a hybrid direct iteration/finite difference procedure is suggested, which would use the converged direct iteration solution of the simple Oswatitsch model as the starting solution for the finite difference procedure.