Initial-value problems: multi-step methods

In this chapter, the discretization of continuous systems is presented. The explicit and implicit discrete maps are discussed for numerical predictions of continuous systems. Basic discrete schemes are presented which include forward and backward Euler methods, midpoint, and trapezoidal rule method. An introduction to Runge–Kutta methods is presented, and the Taylor series method and second-order Runge–Kutta method are introduced. The explicit Runge–Kutta methods for third and fourth order are systematically presented. The implicit Runge–Kutta methods are discussed based on the polynomial interpolation, which include a generalized implicit Runge–Kutta method, Gauss method, Radau method, and Lotta methods. In addition to one-step methods, implicit and explicit multi-step methods are discussed, including Adams–Bashforth method, Adams–Moulton methods, and explicit and implicit Adams methods.

The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.

This paper is aimed at analyzing the consistency and convergence of a derived explicit fourth-stage fourth-order Runge-Kutta method. The analysis revealed that the method is consistent and convergent. The implementation of this method on initial-value problems was done in previous paper, and it revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method.

Inverse linear multistep methods (ILMMs) for first and second order differential equations have been proved to be suitable numerical methods for the solution of inverse initial value problems (IVPs). This paper presents the hybrid version of the ILMMs for the numerical solution of second order inverse IVPs. The stability of the proposed methods is represented in the boundary locus graph. The applicability of the schemes is demonstrated herein for the solution of linear and nonlinear problems. Computational results on the problems are compared with those from the existing method and ode45 (the explicit Runge–Kutta method).

An optimized explicit Runge–Kutta–Nyström method for the numerical solution of orbital and related periodical initial value problems

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

Higher-order Runge-Kutta (RK) algorithms employing local truncation error (LTE) estimates have had very limited success in solving stiff differential equations. These LTEs do not recognize stiffness until the region of instability has been crossed after which no correction is possible. A new technique has been designed, using the local stiffness function (LSF), which can detect stiffness very early before instability occurs. The LSF is a normalized dimensionless ratio which is essentially based on the product of the step size and the geometric mean of all the slopes. It is exceedingly sensitive to the onset of stiffness. Together, the LSF and the LTE form a complementary pair which can cooperate to help solve some mildly stiff equations which were previously intractable to RK algorithms alone. Examples are given of implementation and LSF performance.

In this paper we study a class of explicit pseudo two-step Runge-Kutta (EPTRK) methods for first-order ODEs for parallel computers. We investigate linear stability and derive methods with enlarged stability regions. In numerical experiments on a shared memory computer we compare a parallel variable step size EPTRK implementation with the effcient sequential Runge-Kutta method dopri5.

Parallel iteration across the steps of high-order Runge-Kutta methods for nonstiff initial value problems

When two or more species compete with each other to gain the same limited resources, this situation is known as competition interaction or Competitive Hunters Model. In this study, we plot exact orbit and determine the pattern of behavior by using the Volterra mapping. We also test the accuracy of our numerical approximation methods. Selected numerical methods are Euler, Taylor Series and Runge-Kutta method to plot the orbit for this competitive model. The accuracy of determining the precise orbit and behavior patterns for Competitive Hunters Model for Runge-Kutta, Euler and Taylor Series methods will be studied. Exact orbit from Volterra mapping will be used as the control factor. Observation extended by taking data from experiment performed on P. Caudatum and P. Aurelia samples to observe the theory of competition between this two species when they interact. Base on competitive exclusion principle; exclusion often occurs at a slow rate, therefore it was difficult to determine if equilibrium and stability exist, however it depends on their initial condition and initial numbers of sample for those population that compete with each other. As a solution, we use numerical approximation methods and carrying capacity that we have obtained from curve fitting process as a predictive tool to determine which species will dominate the other species in this competitive hunters' situation.

Parallel implementation for the two-stage SDIRK methods via diagonalization

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations of transient problems can be described by the first-order ordinary differential equations, in which the right-hand side is decomposed into two parts, a linear part and a nonlinear part. In the proposed methods of different orders, the responses of the linear part at the previous step are transferred by the generalized Padé approximations, and the nonlinear part’s responses of the previous step are approximated by the Gauss–Legendre quadrature together with the explicit Runge–Kutta method, where the explicit Runge–Kutta method is used to calculate function values at quadrature points. For reducing computations and rounding errors, the 2m algorithm and the method of storing an incremental matrix are employed in the calculation of the generalized Padé approximations. The proposed methods can achieve higher-order accuracy, unconditional stability, flexible dissipation, and zero-order overshoots. For linear transient problems, the accuracy of the proposed methods can reach 10−16 (computer precision), and they enjoy advantages both in accuracy and efficiency compared with some well-known explicit Runge–Kutta methods, linear multi-step methods, and composite methods in solving nonlinear problems.

Chapter 8 Numerical analysis of dynamical systems

Stability of some continuously imbedded Runge-Kutta methods of Sarafyan

The problem is to calculate an approximate solution of an initial value problem for an autonomous system of N ordinary differential equations.Using fast power series techniques, we exhibit an algorithm for the pth-order Taylor series method re-N quiring only 0(p In p) arithmetic operations per step as p -> + °°.(Moreover, we Nshow that any such algorithm requires at least 0(p ) operations per step.)We compute the order which minimizes the complexity bounds for Taylor series and linearRunge-Kutta methods and show that in all cases this optimal order increases as the error criterion e decreases, tending to infinity as e tends to zero.Finally, we show that if certain derivatives are easy to evaluate, then Taylor series methods are asymptotically better than linear Runge-Kutta methods for problems of small dimension N.