Explicit Difference Scheme Research Articles

Modelling transient heat and moisture transfer in unsaturated soil using a parallel computing approach

AbstractA parallel numerical model, employing a finite difference explicit scheme for the analysis of coupled heat and moisture transfer in unsaturated soil, is employed to simulate a laboratory experiment of heating of medium sand. The model, written in a two‐dimensional polar co‐ordinate formulation, is programmed in the concurrent language Occam and executed on a parallel computing network of transputers. Parallelization is adopted as a means of overcoming computing difficulties, which limited numerical solutions to those at steady state, to enable transient behaviour to be simulated.The parallel algorithm was found to be very efficient, enabling a full solution of transient behaviour to be obtained. An investigation of the ability of the model to accurately simulate the complex, interrelated coupled nature of both two‐dimensional transient and steady‐state behaviour yielded very good correlation between experimental and numerical results. It can therefore be concluded that overall the results obtained provide confidence in the validity of the approach proposed.

A modification of the Petrov–Galerkin method for the transient convection–diffusion equation

AbstractA variation of the Petrov–Galerkin method of solution of a partial differential equation is presented in which the weight function applied to the time derivative term of the transient convection–diffusion equation is different from the weight function applied to the special derivatives. This allows for the formulation of fourth‐order explicit and centred difference schemes. Comparison with analytic solutions show that these methods are able to capture steep wave fronts. The ability of the explicit method to capture wave fronts increases as the amount of convective transport increases.

Sensitivity analysis of the non‐linear dynamic viscoplastic response of 2‐D structures with respect to material parameters

AbstractA computational procedure is presented for evaluating the sensitivity coefficients of the viscoplastic response of structures subjected to dynamic loading. A state of plane stress is assumed to exist in the structure, a velocity strain‐Cauchy stress formulation is used, and the geometric non‐linearities arising from large strains are incorporated. The Jaumann rate is used as a frame indifferent stress rate. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. The equations of motion emanating from a finite element semi‐discretization are integrated using an explicit central difference scheme with an implicit stress update. The sensitivity coefficients are evaluated using a direct differentiation approach. Since the domain of integration is the current configuration, the sensitivity coefficients of the spatial derivatives of the shape functions must be included. Numerical results are presented for a thin plate with a central circular cutout subjected to an in‐plane compressive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients, and spatial distributions at selected times are presented.

Numerical tests of a refined flux corrected transport advection scheme

In the last years several high resolution schemes for solving the advection equation have been developed. One group of these second order, oscillation free, explicit scalar difference schemes are the Flux Corrected Transport (FCT) algorithms. Within these algorithms the flux is computed as a weighted combination of fluxes from a monotonic low order scheme and a higher order scheme. In the construction of FCT algorithms there are three components of interest: the low order scheme, the flux-limiting algorithm and the high order scheme. In this paper the often used Lax Wendroff high order scheme is replaced by the Adams Bashforth scheme. Results of different well known two dimensional tests are shown. Also two new three dimensional test problems with nonzero topography, which are closer to meteorological applications, are introduced and applied to the proposed scheme and to several other advection schemes.

Explicit energy conservative difference schemes for nonlinear dynamical systems with at most quartic potentials

We report a conservative difference scheme for nonlinear dynamical systems with energy of the form H = ( du dt ) 2 2+V(u) . If the potential V(u) is a polynomial of degree four or less, then the scheme is explicit and identical with McMillan's integrable mapping of the plane. We extend the method to systems of several variables. We also discuss whether or not the schemes are symplectic.

Application of symplectic algorithm to dynamical astronomy. III

The two previous papers of this series [1,2] qualitatively clarified the main function of symplectic algorithm [3–6] of maintaining the global structure of hamiltonian systems. In this paper we exploit one of its quantitative advantages—the control of rapid growth of along-track error. With a small improvement in the explicit symplectic difference scheme for separable systems, the algorithm can be extended to more general systems containing small dissipative and non-separable terms. Very good results were obtained showing that the symplectic algorithm has great advantages over usual, non-symplectic ones.

An improved splitting method

In this paper, an improved splitting method, based on the completely square-conservative explicit difference schemes, is established. Not only can the time-direction precision of this method be higher than that of the traditional splitting methods but also can the physical feature of mutual dependence of the fast and the slow stages that are calculated separately and splittingly be kept as well. Moreover, the method owns an universality, it can be generalized to other square-conservative difference schemes, such as the implicit and complete ones and the explicit and instantaneous ones. Good time benefits can be acquired when it is applied in the numerical simulations of the monthly mean currents of the South China Sea.

Investigation of economical difference systems for dynamical problems in elasticity theory in a class of nonsmooth solutions

Using an explicit central difference scheme, an economical difference system is constructed for the first boundary-value problem of the dynamical theory of elasticity for velocities and stresses. Estimates of the rate of convergence of the difference scheme is obtained under weak hypotheses on the smoothness of solutions to the differential problem.

Nonlinear explicit difference schemes for solving parabolic equations

A method is proposed for solving initial-boundary-value problems for parabolic equations by means of reducing them to Cauchy problems for systems of ordinary differential equations and applying to the latter nonlinear explicit numerical methods.

Order conditions of two kinds of canonical difference schemes

The main purpose of this paper is to develop and simplify the general conditions for an s-stage explicit canonical difference scheme to be of q th order while the simplified order conditions for canonical RKN methods, which are applied to special kinds of second order ordinary differential equations, are also obtained here.

Four-point explicit difference schemes for the dispersive equation

A class of three-level explicit difference schemes for the dispersive equation u1=auzzz are established. These schemes have higher stability and involve four mesh points at the middle level. Their local truncation errors are O(τ+h) and stability conditions are from |R|≤0.25 to |R|≤10, where |R|=|a|τ/h3, which, is much better than |R|≤0.25[1].

Explicit Energy-Conserving Difference Schemes for Quartic Nonlinear Hamiltonian Systems

Explicit Energy-Conserving Difference Schemes for Quartic Nonlinear Hamiltonian Systems

Numerical investigation of three-dimensional shock focusing effects by an invariant difference scheme

Symmetry preserving difference schemes which are found on the basis of the Lie-group theory for partial differential equations applied to their first differential approximations are used to simulate the evolution and reflection of shock waves in a cubical cavity with rigid walls and walls with outflow boundary conditions. The explicit difference scheme was used for a shock capturing technique to find and trace the discontinuities in the flow field. The numerical model is based on the strong conservative form of the three-dimensional time-dependent Euler equations with an equation of state for an ideal gas. The three-dimensional time-dependent shock waves which are created by a centered gas volume under high pressure interact with the plane walls of the cavity and lead to focusing effects after the explosion. To follow the propagation of strong plane or spherical shock waves the shock capturing method was proved to be sufficient concerning the spatial resolution of the discontinuities. When interacting with the rigid walls of the cavity the shock waves are assumed to be reflected specularly and to interact with shock waves and contact discontinuities moving from the center to the edges and corners of the cavity. The symmetry preserving character of the invariant difference scheme under use was proved numerically by calculations over long time. No significant deviation of the symmetry character of the imposed initial distribution was found. Additionally, a numerical analysis of the conservation of energy in the cube was performed to find the behavior of the energy in time.

Effect of cross-sectional warping of anisotropic sandwich laminates due to dynamic loads using a refined theory and C° finite elements

An attempt has been made to study the effect of cross-sectional warping in the symmetrically laminated anisotropic composite sandwich plates for transient loads. A higher-order shear deformation theory (HOST) is used in conjunction with the simple displacement based C° finite element method (FEM). As is well-known, the classical first-order theories hitherto considered were inadequate to describe the propagation of waves in the highly orthotropic sandwich laminates. The present theory, which is more accurate than the Reissner-Mindlin theory, is applied herein, for the evaluation of plate response to different types of dynamic loads. An explicit central difference scheme is employed for the integration of dynamic equations of equilibrium with a diagonalized mass matrix obtained by a special procedure applicable to quadrilateral isoparametric elements. The numerical results of the present investigation have been compared with the first-order shear deformation theory (FOST) and the differences between HOST and FOST are examined. The results presented here should be useful in obtaining better correlation between theory and experiment, and to numerical analysts in verifying their results.

A beam-propagation technique to analyze integrated photonic circuits

An explicit central difference scheme to implement the beam-propagation algorithm for analyzing pulse propagation in photonic integrated circuits is discussed. The method has been successfully applied to planar waveguides, channel waveguides, and waveguide couplers. The method shows stability even after long propagation distances and does not involve computation of fast Fourier transform at any step, contrary to the methods discussed earlier. © 1992 John Wiley & Sons, Inc.

Prediction of dynamic response of blast-loaded reinforced concrete structures

This paper is concerned with the nonlinear analysis of plane and axisymmetric reinforced concrete structures subjected to transient impulsive loading. A rate and history dependent constitutive model for concrete is developed in which the compressive behaviour is modelled as an elasto-viscoplastic material and in tension as a linear elastic strain softening material. Viscoplastic response is governed by two rate-dependent bounding loading surfaces combined with rate-sensitive hardening, softening and flow rules. Tensile cracking is controlled by a rate-dependent crack initiation criterion and a fracture energy based nonlinear tension softening rule. Steel is modelled as a strain rate-dependent uniaxial elasto-viscoplastic material. The equations of motion are derived using the principle of virtual displacements in the total Lagrangian approach. Isoparametric elements with special embedded axial members are used to discretize concrete and steel in space. An explicit central difference scheme is adopted for time integration of equations of motion. A computer program, FEABRS, written for linear and nonlinear dynamic analysis is briefly described. As an illustration, an impulsively loaded concrete beam is analysed and the results are compared with data obtained elsewhere. Finally, a parametric study is presented.

An Unconditionally Stable Three-Level Explicit Difference Scheme for the Schrödinger Equation with a Variable Coefficient

A three-level explicit difference scheme for the Schrödinger equation with a variable coefficient is established. It is proved by the discrete energy method that the scheme is unconditionally stable.MSC codes65M0665M12KeywordsSchrödinger’s equationfinite-difference schemestabilitydiscrete energy method

On spurious asymptotic numerical solutions of explicit Runge-Kutta methods

The bifurcation diagram associated with the logistic equation υn+1 = aυn(1 − υn) is by now well known, as is its equivalence to solving the ordinary differential equation (ODE) u′ = αu(1 − u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. We investigate, both analytically and computationally, Runge-Kutta schemes applied to the equation u′=f(u), for f(u) = αu{1 − u) and f(u) = au(1 − u)(b − u), contrasting their behaviour with the explicit Euler scheme. We determine and provide a local analysis of bifurcations to spurious fixed points and periodic orbits. In particular we show that these may appear below the linearised stability limit of the scheme, and may consequently lead to erroneous computational results.

A Fully-Discrete Spectral Method for Delay-Differential Equations

In this paper a new Lanczos-tau method for solving linear functional differential equations is introduced. The scheme has infinite order of accuracy both in time and in the delayed argument. The high accuracy in time is obtained without increasing the computational work and memory space which is needed for a one-step explicit difference scheme. The article demonstrates how to implement the algorithm in a robust and efficient manner and to treat problems with piecewise continuous initial function. Numerical results illustrating the behavior of the method when faced with difficult problems are presented and the numerical results are compared to those obtained by using two other methods.

Non-linear finite element dynamic analysis of two-dimensional concrete structures

This paper presents a numerical procedure for predicting the non-linear dynamic response of plane and axisymmetric reinforced concrete structures. Isoparametric elements with special embedded axial members are used to discretize concrete and steel in space. A summary of a rate and history dependent constitutive model for progressive failure analysis of concrete is given in which the compression behaviour is modelled as a strain rate sensitive elasto-viscoplastic material and in tension as strain rate dependent linear elastic strain softening material. The different rales governing the pre-failure and post-failure behaviour in compression and tension are developed in which the strain rate dependency is included. Steel is modelled as a strain rate dependent uniaxial elasto-viscoplastic material. Explicit central difference scheme in conjunction with an energy balance check is employed for time integration of equations of motion. A computer program for linear and non-linear dynamic analysis of concrete structures is described. Finally, some numerical applications are presented.