Explicit Hybrid Method Research Articles

EXPLICIT FIFTH-ORDER HYBRID METHODS FOR DYNAMICAL SYSTEMS WITH DELAY

In the paper we consider the systems of delay differential equations, which are mathematical models of many technical processes with delay in time. An explicit fifth-order hybrid method for systems of delay differential equations for the variable step of numerical integration is constructed. This algorithm is based on the most commonly used explicit fifth-order Runge-Kutta methods for ordinary systems of differential equations and the construction of Newton polynomials for the prehistory of the model and Taylor formula. The basic principles of construction of such explicit hybrid methods of higher orders of convergence are given. An accurate estimate of the local error of numerical integration by this method is obtained.
 This algorithm allows the use of numerical integration steps greater than the magnitude of the delay or procedures for adjusting the magnitude of the step depending on the calculation error. Such a problem of numerical integration of time-delay dynamical systems occurs when the time interval is large enough compared to the delay. Implicit continuous Runge-Kutta methods were previously used for numerical integration such problems, which complicated the numerical algorithm, because nonlinear systems of equations have to be solved at each step of numerical integration. The explicit hybrid method constructed by us for systems of delay differential equations is convenient for programming, has a high speed of calculation of numerical solution, in comparison with implicit methods for such models of problems. Also, this method has no restrictions on the magnitude of the delay, in contrast to the hybrid methods with decomposition in Taylor series for delay. The method obtained by us can be used to construct maps of dynamic modes in the study of regular and chaotic behavior of time-delay dynamical systems.

Modified Hybrid Method with Four Stages for Second Order Ordinary Differential Equations

A modified explicit hybrid method with four stages is presented, with the first stage exactly integrating exp(wx), while the remaining stages exactly integrate sin(wx) and cos(wx). Special attention is paid to the phase properties of the method during the process of parameter selection. Numerical comparisons of the proposed and existing hybrid methods for several second-order problems show that the proposed method gives high accuracy in solving the Duffing equation and Kramarz’s system.

An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation

We extend the explicit hybrid numerical method for solving the Allen–Cahn (AC) equation to the scheme for the nonlocal AC equation with isotropically symmetric interfacial energy. The proposed method combines the previous explicit hybrid method with a space-time dependent Lagrange multiplier which enforces conservation of mass. We perform numerical tests for the area-preserving mean curvature flow, which is the basic property of the nonlocal AC equation. The numerical results show good agreement with the theoretical solutions. Furthermore, to demonstrate the usefulness of the proposed method, we perform a cell growth simulation in a complex domain. Because the proposed numerical scheme is explicit, it is remarkably simple to implement the numerical solution algorithm on complex discrete domains.

An Accurate and Practical Explicit Hybrid Method for the Chan–Vese Image Segmentation Model

In this paper, we propose a computationally fast and accurate explicit hybrid method for image segmentation. By using a gradient flow, the governing equation is derived from a phase-field model to minimize the Chan–Vese functional for image segmentation. The resulting governing equation is the Allen–Cahn equation with a nonlinear fidelity term. We numerically solve the equation by employing an operator splitting method. We use two closed-form solutions and one explicit Euler’s method, which has a mild time step constraint. However, the proposed scheme has the merits of simplicity and versatility for arbitrary computational domains. We present computational experiments demonstrating the efficiency of the proposed method on real and synthetic images.

A Two-Step Modified Explicit Hybrid Method with Step-Size-Dependent Parameters for Oscillatory Problems

A new two-step modified explicit hybrid method with parameters depending on the step-size is constructed. This method is derived using the coefficients from a sixth-order explicit hybrid method with extended interval of absolute stability and then imposed each stage of the modified formula to exactly integrate the differential equations with solutions that can be expressed as linear combinations of sinwx and coswx, where w is the known frequency. Numerical results show the advantage of the new method for solving oscillatory problems.

Continuous Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations

This paper presents an explicit hybrid method for direct approximation of second order ordinary differential equations. The approach adopted in this work is by interpolation and collocation of a basis function and its corresponding differential system respectively. Interpolation of the basis function was done at both grid and off-grid points while the differential systems are collocated at selected points. Substitution of the unknown parameters into the basis function and simplification of the resulting equation produced the required continuous, consistent and symmetric explicit hybrid method. Attempts were made to derive starting values of the same order with the methods using Taylor’s series expansion to circumvent the inherent disadvantage of starting values of lower order. The methods were applied to solve linear, non-linear, Duffing equation and a system of equation second-order initial value problems directly. Errors in the results obtained were compared with those of the existing implicit methods of the same and even of higher order. The comparison shows that the accuracy of the new method is better than the existing methods.

Solving Oscillatory Delay Differential Equations Using Block Hybrid Methods

A set of order condition for block explicit hybrid method up to order five is presented and, based on the order conditions, two-point block explicit hybrid method of order five for the approximation of special second order delay differential equations is derived. The method is then trigonometrically fitted and used to integrate second-order delay differential equations with oscillatory solutions. The efficiency curves based on the log of maximum errors versus the CPU time taken to do the integration are plotted, which clearly demonstrated the superiority of the trigonometrically fitted block hybrid method.

Block Hybrid Method with Trigonometric-Fitting for Solving Oscillatory Problems

In this paper, we develop algebraic order conditions for two-point block hybrid method up to order five using the approach of B-series. Based on the order conditions, we derive fifth order two-point block explicit hybrid method for solving special second order ordinary differential equations (ODEs), where the existing explicit hybrid method of order five is used to be the method at the first point. The method is then trigonometrically fitted so that it can be suitable for solving highly oscillatory problems arising from special second order ODEs. The new trigonometrically-fitted block method is tested using a set of oscillatory problems over a very large interval. Numerical results clearly showed the superiority of the method in terms of accuracy and execution time compared to other existing methods in the scientific literature.

A pair of explicit hybrid methods with extended interval of periodicity for second order initial value problems

A pair of explicit hybrid methods with extended interval of periodicity for second order initial value problems

Multi-step hybrid methods for special second-order differential equations y ″(t) = f(t,y(t))

In this paper, multi-step hybrid methods for solving special second-order differential equations yź(t) = f(t,y(t)) are presented and studied. The new methods inherit the frameworks of RKN methods and linear multi-step methods and include two-step hybrid methods proposed by Coleman (IMA J. Numer. Anal. 23, 197---220, 8) as special cases. The order conditions of the methods were derived by using the SN-series defined on the set SNT of SN-trees. Based on the order conditions, we construct two explicit four-step hybrid methods, which are convergent of order six and seven, respectively. Numerical results show that our new methods are more efficient in comparison with the well-known high quality methods proposed in the scientific literature.

An explicit hybrid method for multi-term fractional differential equations based on Adams and Runge-Kutta schemes

An explicit hybrid numerical algorithm is proposed by combining Adams-type and Runge-Kutta (RK) methods to solve fractional differential equations (FDEs). A general kind of multi-term FDEs is transformed into combined systems containing both FDEs and ordinary differential equations (ODEs). An explicit Adams-type scheme is constructed to integrate the FDEs. With the obtained solutions of FDEs, the ODEs are explicitly solved by the classical fourth-order RK scheme. Several types of FDEs are presented to validate the proposed method. Compared with an extended predictor corrector method, the presented method can provide more accurate results by an order of magnitude at the expense of one half of computational resources. With such high precision and efficiency, the presented method is applied in numerical simulations of long-term dynamic responses such as asymptotic periodic or limit cycle solutions of fractional dynamic systems.

Zero-dissipative phase-fitted hybrid methods for solving oscillatory second order ordinary differential equations

In this paper, zero-dissipative phase-fitted two-step hybrid methods are developed for the integration of second-order periodic initial value problems. The phase-fitted hybrid methods are constructed using similar approaches introduced by Papadopoulos et al. [1]. This new methods are based on the existing explicit hybrid methods of order four and six. Numerical illustrations indicate that the new methods are much more efficient than the existing methods.

Parallel Implementation of Block Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations

Parallel Implementation of Block Explicit Hybrid Method for Solving Second Order Ordinary Differential Equations

Application of the Hybrid Differential Transform Method to the Nonlinear Equations

In this paper, a hybrid method is introduced briefly to predict the behavior of the non-linear partial differential equations. The method is hybrid in the sense that different numerical methods, differential transform and finite differences, are used in different subdomains. Our aim of this approach is to combine the flexibility of differential transform and the efficiency of finite differences. An explicit hybrid method for the transient response of inhomogeneous nonlinear partial differential equations is presented; applying finite difference scheme on the fixed grid size is used to approximate the space discretisation, whereas the differential transform method is used for time operator. Comparison of the efficiency of the different approaches is a very important aspect of this study. In our test cases, the hybrid approach is faster than the corresponding highly optimized finite difference method in two dimensional computations. We compared our hybrid approach’s results with the exact and/or numerical solutions of PDE which obtained from Adomian Decomposition Method. Results show that the hybrid approach may be an important tool to reduce the execution time and memory requirements for large scale computations and get remarkable results in predicting the solutions of nonlinear initial value problems.

An Embedded Explicit Hybrid Method for Ordinary Differential Equations

An Embedded Explicit Hybrid Method for Ordinary Differential Equations

A Variable Step‐Size Exponentially Fitted Explicit Hybrid Method for Solving Oscillatory Problems

An exponentially fitted explicit hybrid method for solving oscillatory problems is obtained. This method has four stages. The first three stages of the method integrate exactly differential systems whose solutions can be expressed as linear combinations of {1, x, exp(μx), exp(−μx)}, μ ∈ C, while the last stage of this method integrates exactly systems whose solutions are linear combinations of {1, x, x2, x3, x4, exp(μx), exp(−μx)}. This method is implemented in variable step‐size code basing on an embedding approach. The stability analysis is given. Numerical experiments that have been carried out show the efficiency of our method.

Strong stability preserving hybrid methods

This paper is concerned with the strong stability preserving (SSP) time discretizations for semi-discrete systems, obtained from applying the method of lines to time-dependent partial differential equations. We focus on the construction of explicit hybrid methods with nonnegative coefficients, which are a class of multistep methods incorporating a function evaluation at an off-step point. A series of new SSP methods are found. Among them, the low order methods are more efficient than some well known SSP Runge–Kutta or linear multistep methods. In particular, we present some fifth to seventh order methods with nonnegative coefficients, which have healthy CFL coefficients. Finally, some numerical experiments on the Burgers equation are given.

A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems

In this paper we present a kind of trigonometrically fitted explicit two-step hybrid method which achieves algebraic order six. The new method is zero dissipative, phase fitted, and almost P-stable. Numerical experiments from its application to well-known periodic orbital problems show that the new method is more competitive when comparing with the codes proposed by Franco [J.M. Franco, A class of explicit two-step hybrid methods for second-order IVPs, J. Comput. Appl. Math. 187 (2006) 41–57], Anastassi and Simos [Z.A. Anastassi, T.E. Simos, A trigonometrically fitted Runge–Kutta method for the numerical solution of orbital problems, New. Astron. 10 (2005) 301–309].

An adapted explicit hybrid method of Numerov-type for the numerical integration of perturbed oscillators

In 1971, Scheifele has designed a method for the integration of perturbed oscillators by refining the classical method of power series. In this paper, we examine how we can extend the algorithm of Scheifele for the construction of a fourth-order hybrid method without explicit first derivatives. An application to some popular test problems such as the satellite problem shows the properties of the developed algorithm.

A class of explicit two-step hybrid methods for second-order IVPs

A class of explicit two-step hybrid methods for the numerical solution of second-order IVPs is presented. These methods require a reduced number of stages per step in comparison with other hybrid methods proposed in the scientific literature. New explicit hybrid methods which reach up to order five and six with only three and four stages per step, respectively, and which have optimized the error constants, are constructed. The numerical experiments carried out show the efficiency of our explicit hybrid methods when they are compared with classical Runge–Kutta–Nyström methods and other explicit hybrid codes proposed in the scientific literature.