Partitioned adaptive Runge-Kutta methods and their stability

Summary This paper deals with the solution of partitioned systems of nonlinear stiff differential equations. Given a differential system, the user may specify some equations to be stiff and others to be nonstiff. For the numerical solution of such a system partitioned adaptive Runge-Kutta methods are studied. Nonstiff equations are integrated by an explicit Runge-Kutta method while an adaptive Runge-Kutta method is used for the stiff part of the system.The paper discusses numerical stability and contractivity as well as the implementation and usage of such compound methods. Test results for three partitioned stiff initial value problems for different tolerances are presented.     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

Quadratic invariants and multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

In this paper, we study the preservation of quadratic conservation laws of Runge-Kutta methods and partitioned Runge-Kutta methods for Hamiltonian PDEs and establish the relation between multi-symplecticity of Runge-Kutta method and its quadratic conservation laws. For Schrödinger equations and Dirac equations, it reveals that multi-symplectic Runge-Kutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation, respectively.     

Application of the quasi-spectral fourier method to soliton equations

A numerical approach combining the quasi-spectral Fourier method and the Runge-Kutta technique is proposed for the numerical study of the long wavelength regularized equation and the Camassa-Holm and Holm-Hone equations. Test results are presented for soliton and peakon solutions.     

A New Integrator for Special Third Order Differential Equations With Application to Thin Film Flow Problem

In recent time, Runge-Kutta methods that integrate special third order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Albeit, the methods require evaluation of three set of equations to proceed with the numerical integration. In this paper, we propose a class of multistep-like Runge-Kutta methods (hybrid methods), which integrates special third order ODEs directly. The method is completely derivative-free. Algebraic order conditions of the method are derived. Using the order conditions, a four-stage method is presented. Numerical experiment is conducted on some test problems. The method is also applied to a practical problem in Physics and engineering to ascertain its validity. Results from the experiment show that the new method is more accurate and efficient than the classical Runge-Kutta methods and a class of direct Runge-Kutta methods recently designed for special third order ODEs.     

Numerical solutions and stability analysis for solitary waves of complex modified Korteweg–de Vries equation using Chebyshev pseudospectral methods

In this research article, the authors investigate the interaction of solitary waves for complex modified Korteweg–de Vries (CMKdV) equations using Chebyshev pseudospectral methods. The proposed method is established in both time and space to approximate the solutions and to prove the stability analysis for the equations. The derivative matrices are defined at Chebyshev–Gauss–Lobbato points and the problem is reduced to a diagonally block system of coupled nonlinear equations. For numerical experiments, the method is tested on a number of different examples to study the behavior of interaction of two and more than two solitary waves, single solitary wave at different amplitude parameters and different polarization angles. Numerical results support the theoretical results. A comprehensive comparison of numerical results with the exact solutions and other numerical methods are presented. The rate of convergence of the proposed method is obtained up to seventh-order.     

Numerical Stability and Oscillations of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments of Advanced Type

For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.     

A new Chebyshev polynomial approximation for solving delay differential equations

The purpose of this study is to give a Chebyshev polynomial approximation for the solution of mth-order linear delay differential equations with variable coefficients under the mixed conditions. For this purpose, a new Chebyshev collocation method is introduced. This method is based on taking the truncated Chebyshev expansion of the function in the delay differential equations. Hence, the resulting matrix equation can be solved, and the unknown Chebyshev coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.     

Half-explicit Runge-Kutta methods for semi-explicit differential-algebraic equations of index 1

Summary For the numerical solution of non-stiff semi-explicit differentialalgebraic equations (DAEs) of index 1 half-explicit Runge-Kutta methods (HERK) are considered that combine an explicit Runge-Kutta method for the differential part with a simplified Newton method for the (approximate) solution of the algebraic part of the DAE. Two principles for the choice of the initial guesses and the number of Newton steps at each stage are given that allow to construct HERK of the same order as the underlying explicit Runge-Kutta method. Numerical tests illustrate the efficiency of these methods.     

Iterative solutions of nonlinear initial value differential equations in Chebyshev series using Lie series

Summary In this paper Lie series are presented in Chebyshev form and applied to the iterative solution of initial value problems in differential equations. The resulting method, though algebraically complicated, is of theoretical interest as a generalisation of Taylor series methods and iterative Chebyshev methods. The theory of the method is discussed and the solutions of some simple scalar equations are analysed to illustrate the behaviour of the process.     

Partitioned adaptive Runge-Kutta methods for the solution of nonstiff and stiff systems

Summary For the numerical solution of initial value problems of ordinary differential equations partitioned adaptive Runge-Kutta methods are studied. These methods consist of an adaptive Runge-Kutta methods for the treatment of a stiff system and a corresponding explicit Runge-Kutta method for a nonstiff system. First we modify the theory of Butcher series for partitioned adaptive Runge-Kutta methods. We show that for any explicit Runge-Kutta method there exists a translation invariant partitoned adaptive Runge-Kutta method of the same order. Secondly we derive a special translaton invariant partitioned adaptive Runge-Kutta method of order 3. An automatic stiffness detection and a stepsize control basing on Richardson-extrapolation are performed. Extensive tests and comparisons with the partitioned RKF4RW-algorithm from Rentrop [16] and the partitioned algorithm LSODA from Hindmarsh [9] and Petzold [15] show that the partitoned adaptive Runge-Kutta algorithm works reliable and gives good numericals results. Furthermore these tests show that the automatic stiffness detection in this algorithm is effective.     

A local-global version of a stepsize control for Runge-Kutta methods

In this paper we develop a new procedure to control stepsize for Runge-Kutta methods applied to both ordinary differential equations and semi-explicit index 1 differential-algebraic equations. In contrast to the standard approach, the error control mechanism presented here is based on monitoring and controlling both the local and global errors of Runge-Kutta formulas. As a result, Runge-Kutta methods with the local-global stepsize control solve differential or differential-algebraic equations with any prescribed accuracy (up to round-off errors). For implicit Runge-Kutta formulas we give the sufficient number of both full and modified Newton iterations allowing the iterative approximations to be correctly used in the procedure of the local-global stepsize control. In addition, we develop a stable local-global error control mechanism which is applicable for stiff problems. Numerical tests support the theoretical results of the paper.     

Numerical study of Peakons and k-Solitons of the Camassa-Holm and Holm-Hone equations

The spectral Fourier and Runge-Kutta methods are used to study the Camassa-Holm and Holm-Hone equations numerically. Numerical results for problems with initial data leading to the generation and interaction of peakons and k-solitons are discussed.     

刚性多滞量积分微分方程的Runge—Kutta方法

本文研究了求解刚性多滞量积分微分方程的Runge-Kutta方法的非线性稳定性和计算有效性.经典Runge—Kutta方法连同复合求积公式和Pouzet求积公式被改造用于求解一类刚性多滞量Volterra型积分微分方程.其分析导出了:在适当条件下,扩展的Runge-Kutta方法是渐近稳定和整体稳定的.此外,数值试验表明所给出的方法是高度有效的.     

Generalized Runge-Kutta methods for coupled systems of hyperbolic differential equations

Runge-Kutta formulas are discussed for the integration of systems of differential equations. The parameters of these formulas are square matrices with component-dependent values. The systems considered are supposed to originate from hyperbolic partial differential equations, which are coupled in a special way. In this paper the discussion is concentrated on methods for a class of two coupled systems. For these systems first and second order formulas are presented, whose parameters are diagonal matrices. These formulas are further characterized by their low storage requirements, by a reduction of the computational effort per timestep, and by their relatively large stability interval along the imaginary axis. The new methods are compared with stabilized Runge-Kutta methods having scalar-valued parameters. It turns out that a gain factor of 2 can be obtained.     

High Accuracy Spectral Method for the Space-Fractional Diffusion Equations

In this paper, a high order accurate spectral method is presented for the space-fractional diffusion equations. Based on Fourier spectral method in space and Chebyshev collocation method in time, three high order accuracy schemes are proposed. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency compared with low-order counterparts, and a completely straightforward extension to high spatial dimensions. Some numerical examples, including Allen-Cahn equation, are conducted to verify the effectiveness of this method.     

A CLASS OF TWO-STEP CONTINUITY RUNGE-KUTTA METHODS FOR SOLVING SINGULAR DELAY DIFFERENTIAL EQUATIONS AND ITS STABILITY ANALYSIS

In this paper, a class of two-step continuity Runge-Kutta（TSCRK） methods for solving singular delay differential equations（DDEs） is presented. Analysis of numerical stability of this methods is given. We consider the two distinct cases： （i）τ≥ h, （ii）τ 〈 h, where the delay τ and step size h of the two-step continuity Runge-Kutta methods are both constant. The absolute stability regions of some methods are plotted and numerical examples show the efficiency of the method.     

On the generation of mono-implicit Runge-Kutta-Nyström methods by mono-implicit Runge-Kutta methods

Mono-implicit Runge-Kutta methods can be used to generate implicit Runge-Kutta-Nyström (IRKN) methods for the numerical solution of systems of second-order differential equations. The paper is concerned with the investigation of the conditions to be fulfilled by the mono-implicit Runge-Kutta (MIRK) method in order to generate a mono-implicit Runge-Kutta-Nyström method (MIRKN) that is P-stable. One of the main theoretical results is the property that MIRK methods (in standard form) cannot generate MIRKN methods (in standard form) of order greater than 4. Many examples of MIRKN methods generated by MIRK methods are presented.     

On Chebyshev Solution of Parabolic Partial Differential Equations

Now at Mathemarics Department, Assiut University Egypt A method is presented to transform parabolic equations to asystem of ordinary differential equations for the solution atthe Chebyshev points. The system may be solved analyticallyor by numerical methods and the Chebyshev coefficients are computed.We have the exact solution of a perturbed problem.     

Spectral Method for the Black-Scholes Model of American Options Valuation

In this paper, we devote ourselves to the research of numerical methods for American option pricing problems under the Black-Scholes model. The optimal exercise boundary which satisfies a nonlinear Volterra integral equation is resolved by a high-order collocation method based on graded meshes. For the other spatial domain boundary, an artificial boundary condition is applied to the pricing problem for the effective truncation of the semi-infinite domain. Then, the front-fixing and stretching transformations are employed to change the truncated problem in an irregular domain into a one-dimensional parabolic problem in [−1,1]. The Chebyshev spectral method coupled with fourth-order Runge-Kutta method is proposed for the resulting parabolic problem related to the options. The stability of the semi-discrete numerical method is established for the parabolic problem transformed from the original model. Numerical experiments are conducted to verify the performance of the proposed methods and compare them with some existing methods.