Two-step Implicit Research Articles

Solving First-Order Nonlinear Fuzzy Initial Value Problems Using Two-Step Block Method with Presence of Higher Derivatives

Fuzzy differential equation models are suitable where uncertainty exists for real-world phenomena. Numerical techniques are used to provide an approximate solution to these models in the absence of an exact solution. However, existing studies that have developed numerical techniques for solving FIVPs possess an absolute error accuracy that could be improved. This is as a result of the low order and non-self-starting properties of the developed numerical techniques by previous studies. For this reason, this study, develops an Obrechkoff-type two-step implicit block method with the presence of second and third derivative for the numerical solution of first-order nonlinear fuzzy initial value problems. The convergence properties for the proposed block method are described in detail. Then the proposed method is adopted to solve first-order nonlinear fuzzy initial value problems with triangular and trapezoidal fuzzy numbers. The obtained results indicates that the proposed method effectively solves first-order nonlinear fuzzy initial value problems with better accuracy.

A Filtered Milne-Simpson ODE Method with Enhanced Stability Properties

The Milne-Simpson method is a two-step implicit linear multistep method for the numerical solution of ODEs that obtains the theoretically highest order of convergence for such a method. The stability region of the method is only an interval on the imaginary axis and the method is classified as weakly stable which causes non-physical oscillations to appear in numerical solutions. For this reason, the method is seldom used in applications. This work examines filtering techniques that improve the stability properties of the Milne-Simpson method while retaining its fourth-order convergence rate. The resulting filtered Milne-Simpson method is attractive as a method of lines integrator of linear time-dependent partial differential equations.

Exponentially fitted two-step peer methods for oscillatory problems

This paper concerns the construction of a general class of exponentially fitted two-step implicit peer methods for the numerical integration of Ordinary Differential Equations (ODEs) with oscillatory solution. Exponentially fitted methods are able to exploit a-priori known information about the qualitative behaviour of the solution to efficiently furnish an accurate solution. Moreover, peer methods are very suitable for a parallel implementation, which may be necessary in the discretization of Partial Differential Equations (PDEs) when the number of spatial points increases. Examples of methods with 2 and 3 stages are provided. Numerical experiments are carried out in order to confirm theoretical expectations.

A New Two-step Obrechkoff Method with Vanished Phase-lag and Some of its Derivatives for the Numerical Solution of Radial Schrodinger Equation and Related IVPs with Oscillating Solutions

A new two-step implicit linear Obrechkoff twelfth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schrodinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

Implicit–explicit time discretization coupled with finite element methods for delayed predator–prey competition reaction–diffusion system

The main purpose of this paper is to develop the two-step implicit–explicit(IMEX) time discretization coupled with finite element methods for solving delayed predator–prey competition reaction–diffusion system. Finite element methods are used to discretize the space variables, both IMEX two-step one-leg methods and IMEX linear two-step methods are considered in time discretization, where the nonlinear reaction part is discretized explicitly and the diffusion part is discretized implicitly. The L2-norm error estimates with handling the breaking points are derived. These methods only require solving two independent linear systems with the same coefficient matrix at different time, and avoid solving large-scale systems of nonlinear algebraic equations, thus the resulting schemes are efficient. Some numerical experiments are presented to confirm the theoretical results.

Numerical Solution of First Order Initial Value Problems Using a Self-Starting Implicit Two-Step Obrechkoff-Type Block Method

The conventional two-step implicit Obrechkoff method is a discrete scheme that requires additional starting values when implemented for the numerical solution of first order initial value problems. This paper therefore presents a two-step implicit Obrechkoff-type block method which is self-starting for solving first order initial value problems, hence bypassing the rigour of developing and implementing new starting values for the method. Numerical examples are considered to show the new method performing better when com-pared with previously existing methods in literature.

Space-Time Finite Volume Differencing Framework for Effective Higher-Order Accurate Discretizations of Parabolic Equations

A space-time finite volume differencing method is applied to construct new higher-order one-step and two-step implicit and explicit time integration schemes for parabolic equations. A unified space-time discretization error for approximating the integral form of the equation is first formulated by utilizing general weighted quadratures of neighboring grid points to approximate the flux integrals on space-time control volumes centered about each point. Efficient weighted quadrature approximations of the source term are then sought to account for local space-time flux approximations to all neighboring quadrature points on the control volume through a constrained minimization of the local error. Closed form descriptions of the weights that describe the resulting schemes and the leading coefficients of the residual errors are determined to guide the right selections of time step sizes to ensure consistent higher-order order convergence rates subject to stability requirements. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.

Linearly implicit schemes for a class of dispersive–dissipative systems

We consider initial value problems for semilinear parabolic equations, which possess a dispersive term, nonlocal in general. This dispersive term is not necessarily dominated by the dissipative term. In our numerical schemes, the time discretization is done by linearly implicit schemes. More specifically, we discretize the initial value problem by the implicit---explicit Euler scheme and by the two-step implicit---explicit BDF scheme. In this work, we extend the results in Akrivis et al. (Math. Comput. 67:457---477, 1998; Numer. Math. 82:521---541, 1999), where the dispersive term (if present) was dominated by the dissipative one and was integrated explicitly. We also derive optimal order error estimates. We provide various physically relevant applications of dispersive---dissipative equations and systems fitting in our abstract framework.

A computer code for simulating hyper-concentrated sediment motions in steady and unsteady flows

A numerical model based on the two-phase flow equations is developed for predicting the intense sediment motions in steady and unsteady flows. As the governing equations are coupled and highly nonlinear, the failure of some earlier attempts in applying two-phase flow models to reproduce the laboratory measurements of flow properties has been found to be largely due to the deficiencies in the numerical schemes rather than the parameterisations of physical processes. In this paper, a numerical scheme for the complete set of two-phase flow equations is presented. It is a two-step implicit–explicit finite difference scheme and particularly designed to resolve flow details in the highly concentrated region at a mobile sand bed where vertical gradients of concentration and flow velocities are very large. Two typical coastal/river flow cases were used to verify the model by comparing the computed sediment motions with experimental data. A detailed evaluation of the convergence characteristics, stability and robustness of the scheme is also provided for these test cases. The initial results indicate that the proposed scheme is stable and sufficiently accurate for predicting the fluid and sediment motions under both steady and unsteady conditions.

GP-Stability of two-step implicit runge-kutta methods for delay differential equations

The GP-stability of two-step implicit Runge-Kutta (TIRK) methods for the numerical solution of systems of delay differential equations (DDEs) is considered. We focus on the stability behaviour of TIRK methods in the solution of the linear constant-coefficient systems of DDE with a single delay. We present a sufficient condition of GP-stability of TIRK methods.

Numerical Simulation of Unsteady Nonisothermal Capillary Interfaces

A family of algorithms for simulation of unsteady nonisothermal capillary interfaces has been developed. The algorithms are based on a coordinate transformation method. The time-dependent unknown physical domain is mapped onto a rectangular computational domain, with the explicit form of the mapping function not being known. Four types of temporal discretization are used leading to the first-order accurate one-step implicit method, second-order accurate Crank–Nicolson and trapezoidal methods and second-order accurate two-step implicit method. In all cases, second-order finite-difference approximations were used for spatial discretizations. Various tests demonstrated that the algorithms deliver theoretically predicted accuracy, even for very large interfacial distortions. The Crank–Nicolson and trapezoidal methods have been found to be conditionally stable and thus are not recommended.

Average-State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows

Several new implicit schemes for the solution of the compressible Navier–Stokes equations are presented. These methods are derived from a hierarchy of average-state approximate solutions to the Riemann problem, ranging from the Lax–Friedrichs flux to the exact Riemann-solver flux. In contrast to linearised approximations, these methods will (with certain provisos on the signal velocities) enforce the entropy condition and preserve positivity without the need for additional corrections. The hierarchy also encompasses and explains the origin of many other upwind and centred methods, including the space-time scheme (due to Chang) and the more recent FORCE scheme (due to Toro). Based on an analysis of the above hierarchy, attention is focussed on the development of a new implicit scheme using a positivity-preserving version of Toroet al.'s HLLC scheme, which is the simplest average-state solver capable of exactly preserving isolated shock, contact, and shear waves. Solutions obtained with this method are essentially indistinguishable from those produced with an exact Riemann solver, whilst convergence to the steady state is the most rapid of all the implicit average-stage schemes considered and directly comparable to that of the unmodified Roe scheme. A new two-step implicit method is applied to various test cases, including turbulent flow with shock/boundary-layer interaction. The new time-stepping scheme is composed of two backward Euler steps, but has twice the convergence rate of the backward Euler scheme and alleviates the convergence problems that are often experienced when employing compressive limiter functions.

Depth-Averaged Hydrodynamic Model in Curvilinear Collocated Grid

A two-dimensional (2D) depth-averaged model of turbulent flows has been developed in a boundary-fitted curvilinear coordinate system. Based on the finite-volume method (FVM), the convection terms are discretized by the Roe's scheme of second-order accuracy, as well as the Power-law scheme. The governing equations are solved in a collocated grid system by the efficient fractional two-step implicit algorithm that is regarded superior to the SIMPLEC algorithm. The “time marching” technique has been used to obtain a steady-state solution as a limiting case. The new model has been applied to several problems including pure convection of a sharp step profile, side discharge into an open channel, flow in a meandering channel, and flow in a Parshall flume with supercritical outflow. Comparison with the available data shows that the model is efficient and robust.

Obrechkoff methods having additional parameters for general second-order differential equations

A class of two-step implicit methods involving higher-order derivatives of y for initial value problems of the form y″ = f( t, y, y′) is developed. The methods involve arbitrary parameters p and q, which are determined so that the methods become absolutely stable when applied to the test equation y″ + λy′ + μy = 0. Numerical results for Bessel's and general second-order differential equations are presented to illustrate that the methods are absolutely stable and are of order O( h 4), O( h 6) and O( h 8).

Nonlinear nonparaxial beam-propagation method

An unconditionally unitary beam propagation method for solving the nonlinear nonparaxial TE equation or nonlinear Helmholtz equation is presented for the first time. This method is based on an iterative selfconsistent two-step implicit central difference scheme, which makes it suitable for extension to nonlinear vector wave equations.

An implicit scheme for efficient solution of the coalescence/collision-breakup equation

Detailed moist-convection models of the atmosphere require solution of the coalescence/collision-breakup equation to predict change in the water-drop spectrum. Under certain meteorological conditions, computational stability problems are encountered in attempting to solve this equation by explicit techniques. A stability analysis is performed making use of empirical results that depend on the form of observed drop spectra and the nature of the equation. Information derived from the analysis is used to design a two-step implicit scheme that provides both stability and efficiency.