Block Implicit Methods Research Articles

Implicit stomer cowell k-step hybrid block methods for solution of first order ordinary differential equations (ODEs)

This paper focuses on derivation of two implicit hybrid Block methods at and one method at Each block method produced at yielded uniform order 5 and 7 respectively, while the Block implicit method at k = 4 produced uniform order 9. The block analysis of all the methods derived shows that they are zero stable, consistent and convergent. The superiority of these methods over some existing one were demonstrated numerically.

Blended implicit methods for solving ODE and DAE problems, and their extension for second-order problems

The use of implicit numerical methods is mandatory when solving general stiff ODE/DAE problems. Their use, in turn, requires the solution of a corresponding discrete problem, which is one of the main concerns in the actual implementation of the methods. In this respect, blended implicit methods provide a general framework for the efficient solution of the discrete problems generated by block implicit methods. In this paper, we review the main facts concerning blended implicit methods for the numerical solution of ODE and DAE problems. In addition to this, we study the extension of blended implicit methods for solving second-order problems, which results in a straightforward generalization of the basic theory for such methods. Finally, a few numerical tests obtained with the computational code BiMD, implementing a variable order-variable stepsize blended implicit method, are also reported, in order to confirm the effectiveness of the approach.

Blended implicit methods for the numerical solution of DAE problems

Recently, a new approach for solving the discrete problems, generated by the application of block implicit methods for the numerical solution of initial value problems for ODEs, has been devised [L. Brugnano, Blended block BVMs (B 3 VMs): a family of economical implicit methods for ODEs, J. Comput. Appl. Math. 116 (2000) 41–62; L. Brugnano, C. Magherini, Blended implementation of block implicit methods for ODEs, Appl. Numer. Math. 42 (2002) 29–45; L. Brugnano, D. Trigiante, Block implicit methods for ODEs, in: D. Trigiante (Ed.), Recent Trends in Numerical Analysis, Nova Science Publishers, New York, 2001, pp. 81–105]. This approach is based on the so-called blended implementation of the methods, giving corresponding blended implicit methods. The latter have been implemented in the computational code BiM [L. Brugnano, C. Magherini, The BiM code for the numerical solution of ODEs, J. Comput. Appl. Math. 164–165 (2004) 145–158]. Blended implicit methods are here extended to handle the numerical solution of DAE problems, resulting in a straightforward generalization of the basic approach.

Economical error estimates for block implicit methods for ODEs via deferred correction

Deferred correction is a widely used tool for improving the numerical approximation to the solution of ODE problems [J.R. Cash, WSSIA 2 (1993) 113; J.R. Cash, M.H. Wright, SIAM J. Sci. Statist. Comput. 12 (1991) 971; M. Lentini, V. Pereyra, Math. Comp. 28 (1974) 981; B. Lindberg, BIT 20 (1980) 486; V. Pereyra, Numer. Math. 8 (1966) 376; V. Pereyra, Numer. Math. 10 (1967) 316; H.J. Stetter, Numer. Math. 29 (1978) 425; H.J. Stetter, in: Lecture Notes in Math. vol. 630, Springer, 1978, pp. 245–258; R.D. Skeel, SIAM J. Numer. Anal. 19 (1981) 171; R.D. Skeel, Numer. Math. 48 (1986) 1; P. Zadunaisky, Numer. Math. 27 (1976) 21]. Indeed, it allows to estimate the error due to the use of discrete methods. Such an estimate may be a global one, in the case of continuous BVPs, or a local one, when IVPs are to be approximated [L. Brugnano, in: Lecture Notes in Math., vol. 1196, Springer, 1997, pp. 78–89; L. Brugnano, D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach, 1998]. Recently, it has been implemented in the computational code BiM [L. Brugnano, C. Magherini, J. Comput. Appl. Math. 164–165 (2004) 145, web page: http://math.unifi.it/~brugnano/BiM/index.html] for the numerical solution of stiff ODE-IVPs. In this paper we analyze deferred correction in connection with the methods used in that code, resulting in an overall simplification of the procedure, due to the properties of the underlying methods. The analysis is then extended to more general methods.

Turbulent convection of air-cooled rectangular duct with surface-mounted cross-ribs

Experimental and numerical studies were conducted to investigate the forced convection and flow friction of a turbulent airflow in a horizontal air-cooled rectangular duct, with square-sectioned cross-ribs mounted on its bottom surface. Cross-sectional dimensions of the cross-ribs were 6.37 mm × 6.37 mm. Reynolds number of the fully turbulent flow was maintained constant at 12,380. Heat was supplied uniformly to the airflow via bottom surface of the duct only. Effects of varying the angle formed by the cross-ribs between 30° and 120° on the forced convection and flow friction were studied. It was found that an optimum angle corresponding to the highest heat transfer coefficient occurred between 60° and 70°. Computational predictions of forced convection and flow friction of the same rectangular duct mounted with cross-ribs were performed with the zonal k– ε model, the stability-guaranteed second-order difference scheme and the block implicit method. A comparison with the experimental results indicated that a reasonably good agreement had been achieved. Existence of an optimum angle formed by the cross-ribs between 60° and 70° was also indicated, except the numerical predictions were lower than the experimental findings by approximately 2–10%. However, the flow friction had been slightly over-estimated by 2–11% with the numerical model. It was shown that forced convection could be enhanced sufficiently by mounting cross-ribs on the internal surfaces of a rectangular duct, especially when the optimum angle was used.

Blended implementation of block implicit methods for ODEs

In this paper we further develop a new approach for naturally defining the nonlinear splittings needed for the implementation of block implicit methods for ODEs, which has been considered by Brugnano [J. Comput. Appl. Math. 116 (2000) 41] and by Brugnano and Trigiante [in: Recent Trends in Numerical Analysis, Nova Science, New York, 2000, pp. 81–105]. The basic idea is that of defining the numerical method as the combination ( blending) of two suitable component methods. By carefully choosing such methods, it is shown that very efficient implementations can be obtained. Moreover, some of them, characterized by a diagonal splitting, are well suited for parallel computers. Some numerical tests comparing the performances of the proposed implementation with other existing ones are also presented, in order to make evident the potential of the approach.

Numerical model of a transferred plasma arc

This article presents results of two-dimensional simulations for a transferred plasma arc. Calculations are performed for the internal plasma flow within the torch body, as well as the external plasma jet impinging on a surface. The governing equations describing the plasma flow are self-consistently coupled. These equations are the compressible Navier–Stokes equations for conservation of mass and momentum (in the radial and streamwise directions), conservation of energy, species continuity with finite-rate ionization and recombination kinetics, and the magnetic diffusion equation for the electromagnetics. The unsteady forms of these equations are time marched to steady state using the linearized block implicit method. This model does not employ any adjustable parameters, and therefore enables direct comparison with experiments. Model predictions are compared with experimental measurements of stagnation pressure distributions recorded on a water-cooled copper plate (workpiece), and indicate good agreement, given the lack of adjustable parameters.

A comparative study of pressure correction and block-implicit finite volume algorithms on parallel computers

A comparison of a new parallel block-implicit method and the parallel pressure correction procedure for the solution of the incompressible Navier–Stokes equations is presented. The block-implicit algorithm is based on a pressure equation. The system of non-linear equation s is solved by Newton's method. For the solution of the linear algebraic systems the Bi-CGSTAB algorithm with incomplete lower–upper (ILU) decomposition of the matrix is applied. Domain decomposition serves as a strategy for the parallelization of the algorithms. Different algorithms for the parallel solution of the linear system of algebraic equations in conjunction with the pressure correction procedure are proposed. Three different flows are predicted with the parallel algorithms. Results and efficiency data of the block-implicit method are compared with the parallel version of the pressure correction algorithm. The block-implicit method is characterized by stable convergence behaviour, high numerical efficiency, insensitivity to relaxation parameters and high spatial accuracy. © 1997 John Wiley & Sons, Ltd.

Numerical simulation of a hydrogen arcjet

The two-dimensional, axisymmetric, nonequilibrium ionizing and reacting flow in a hydrogen arcjet thruster is simulated numerically using the linearized block implicit method. A single ionic species (H), single temperature (i.e., equal electron and heavy particle temperatures), and variable properties are used in this formulation to solve the governing equations of mass, species, momentum, and energy. The current distribution is obtained by solving the magnetic diffusion equation at each time step of the time-marching scheme. As an illustration, a hydrogen arcjet having an exit-to-throat area ratio of 25 (nominal 30-kW geometry) is studied. It is found that the species in order of their abundance at the centerline of the exit plane of the thruster consist of atomic hydrogen (H), ions (H), and electrons (e~), and molecular hydrogen (H2), respectively. As expected, near the colder walls of the anode at the exit plane, this abundance shifts in favor of molecular hydrogen, atomic hydrogen, H, and electrons, respectively.

Adaptive block-implicit methods for annular liquid jets

A domain-adaptive technique is used to analyze the dynamic response of inviscid, isothermal, annular liquid jets to velocity fluctuations at the nozzel exit. The adaptive technique maps the unknown, time-dependent, curvilinear geometry of the annular jet into a unit interval and yields a system of integro-differential equations for the mass per unit length, radius, and radial velocity components of the liquid jet. The convergence length is governed by an ordinary differential equation that depends on the values of the dependent variables at the convergence point. The governing equations are solved by means of two methods of lines, which discretize the spatial coordinate but keep continuous the time, and a Newton method. A block-bidiagonal technique is also used to solve the fluid dynamics equations. It is shown that the dynamic response of liquid jets to sinusoidal velocity fluctuations at the nozzle exit is periodic and nearly sinusoidal for small amplitudes of the velocity oscillations at the nozzle exit. For large amplitudes the fluctuations of both the convergence length and the pressure of the gases enclosed by the annular jet are periodic but not sinusoidal. It is also shown that there is a delay time between the velocity fluctuations at the nozzle exit and those of the convergence length. This delay time is nearly independent of the amplitude of the velocity fluctuations at the nozzle exit and decreases as the Weber and Froude numbers and the nozzle exit angle are decreased.

Parallel algorithms for solving initial value problems: front broadening and embedded parallelism

This paper reviews and develops i) broadening of the computation front (BCF) techniques, and ii) block implicit (BI) methods to speed up the numerical solution of IVPS in ODEs by using MIMD computing systems. In particular consideration is given to the performance, accuracy and stability characteristics of the techniques and possible extension of the BCF methods to improve the accuracy. It has been shown that it is not possible to get fourth and higher order parallel explicit Runge-Kutta (RK) methods by using the BCF techniques. This paper also characterizes some of the drawbacks of BCF and considers two further techniques to exploit the parallelism embedded in BI methods.

Dynamic response of liquid curtains to time-dependent pressure fluctuations

A domain-adaptive finite difference method is used to study the dynamic response of incompressible, isothermal, inviscid liquid curtains that are subject to imposed pressure fluctuations. The domain-adaptive method involves a mapping that transforms the time-dependent physical domain into a fixed computational domain and requires that the unknown location of the downstream boundary of the physical domain be determined by means of an ordinary differential equation that ensures global mass conservation. An iterative block-implicit method has been used to solve the finite difference equations, and calculations are presented to illustrate the effects of the Froude number, convergence parameter, nozzle exit angle, amplitude and frequency of the imposed pressure fluctuations, and nozzle exit thickness-to-radius ratio on the liquid curtain convergence length. It is shown that the amplitude of the convergence length oscillations decreases as the Froude number, convergence parameter, frequency of the imposed pressure oscillations, and nozzle exit thickness-to-radius ratio are increased and increases as the nozzle exit angle and amplitude of the imposed pressure fluctuations are increased. The frequency of the convergence length oscillations is only a function of the Froude number and frequency of the imposed pressure fluctuations. It is also shown that the critical value of the pressure coefficient determined from analytical solutions of the steady-state equations can be exceeded without affecting the stability and convergence of the liquid curtain if the frequency of the imposed pressure oscillations is sufficiently high.

A-stable and L-stable block implicit one-step methods with modified algorithm

A class of A-stable and L-stable block implicit one-step methods were studied in Zhou [9]. If the nodes within block were reasonablely chosen, it was shown that the methods had several attractive properties. Here we give a quick review of the methods. Then we discuss the error estimate and step control policy. A C program, ABIOS_4, is given at end of the paper which uses modified algorithm suggested in [9].

Two‐phase laminar axisymmetric jet flow: Explicit, implicit, and split‐operator approximations

AbstractAn hybrid Eulerian‐Lagrangian numerical scheme is developed for a two‐phase problem and four finite‐difference schemes are compared. For this purpose, the problem of hydrodynamic and thermal interactions between a fuel spray and a mixing region of two laminar, unconfined axisymmetric jets is formulated in terms of a set of parabolic differential equations for the gas phase and a set of Lagrangian ordinary differential equations for the condensed phase. Consistent, second‐order accurate hybrid numerical schemes, with the exception of the explicit scheme with an accuracy between linear and quadratic, are used to solve these equations. The subset of gas‐phase equations has been solved by four different numerical methods: a predictor‐corrector explicit method, a sequential implicit method, a block implicit method, and a symmetric operator‐splitting method. The subsystem of liquid‐phase equations is solved along the droplet trajectories by a second‐order Runge‐Kutta scheme. The computations have been made to predict the hydro‐dynamic and thermal mixing regions of the gas phase as well as the trajectories of each individual group of droplets. In addition, the size, velocity and temperature associated with each group are predicted along these trajectories. The relative merits of the above four difference‐schemes are discussed by constructing effectiveness curves. At low error tolerances, the sequential implicit method gives the best results, where for large error tolerances, the explicit and operator splitting give better results. The block implicit scheme is the least effective at all accuracy requirements.

Implicit lambda methods for three-dimensional compressible flow

This paper derives the three dimensional lambda-formulation equations for a general orthogonal curvilinear coordinate system and provides various block-explicit and block-implicit methods for solving them, numerically. Three model problems, characterized by subsonic, supersonic and transonic flow conditions, are used to assess the reliability and compare the efficiency of the proposed methods.

All symmetric interpolatory block-implicit methods of order less than six are A-stable

All symmetric interpolatory block-implicit methods of order less than six are A-stable

Prediction of Cascade Flow Fields Using the Averaged Navier-Stokes Equations

A numerical solution procedure for the ensemble-averaged, compressible, time-dependent Navier-Stokes equations is applied to predict the flow about a cascade of airfoils operating in the transonic flow regime. The equations are solved by the consistently split, linearized block implicit (LBI) method of Briley and McDonald. Boundary conditions are set so as to specify upstream total pressure and downstream static pressure. Turbulence is modeled by a mixing length model. Predictions are made for flow through a compressor cascade configuration. The method yields converged solutions within a relatively small number of time steps ( ≈ 150), which give good comparisons with experimental data.

On the structure and use of linearized block implicit schemes

The recent use of methods which may be termed “linearized block ADI methods” or more generally “consistently split linearized block implicit” methods has been a significant development in the efficient and noniterative solution of certain systems of coupled nonlinear multidimensional partial differential equations. Some observations on their structure, derivation, and use are given. Consistently split linearized block implicit (LBI) methods are unified here and are related to the earlier scalar ADI schemes, as well as to existing iterative and noniterative methods for solving both systems of nonlinear algebraic equations, and systems of nonlinear ordinary differential equations (including those having multipoint boundary conditions). It is shown that the method used by Lindemuth and Killeen and that of Briley and McDonald (utilizing a two-dimensional Crank-Nicolson formulation) are both consistently split block implicit schemes which differ in principle only with regard to implementation of the linearization technique. It is also observed that the first approximate factorization scheme of Beam and Warming utilizes a splitting due to D'Yakanov whose intermediate steps are inconsistent in the sense that they do not approximate the governing equations to within a truncation error which vanishes to some order for small Δt. Methods based on splittings which have inconsistent intermediate steps are placed in a separate category and are shown to present serious difficulties, which apparently have escaped notice, in treating derivative boundary conditions accurately. Although similar difficulties can arise in the transient with consistently split schemes, the consistent splitting normally provides one order of accuracy improvement. It is further demonstrated that the two-level version of the second and more recent “delta” form approximate factorization scheme of Warming and Beam and the earlier method of Briley and McDonald have identical linearized block implicit structures. Finally, further substantial gains in efficiency resulting from reducible block submatrices and the use of multiple time steps are described.

Block Implicit One-Step Methods

A new class of block implicit one-step methods for ordinary differential equations is presented. The methods are based on quadrature and generate function values at nonmesh points through Hermite interpolation. A general convergence theorem for block implicit methods is given, and the stability of the new class of methods is analyzed. The class contains A-stable, stiffly stable, strongly A-stable, and strongly stiffly stable methods. Numerical results demonstrating the efficiency and effectiveness of a particular block method are presented.

Block implicit one-step methods

A new class of block implicit one-step methods for ordinary differential equations is presented. The methods are based on quadrature and generate function values at nonmesh points through Hermite interpolation. A general convergence theorem for block implicit methods is given, and the stability of the new class of methods is analyzed. The class contains A-stable, stiffly stable, strongly A-stable, and strongly stiffly stable methods. Numerical results demonstrating the efficiency and effectiveness of a particular block method are presented.