Symmetric Runge-Kutta Methods Research Articles

Explicit Symmetric Exponential Integrators for Charged-Particle Dynamics in a Strong and Constant Magnetic Field

In this paper, we derive and analyze symmetric exponential integrators for charged-particle dynamics in a strong and constant magnetic field. We first present the scheme of exponential integrators and then establish the symmetry conditions for the methods. Explicit symmetric exponential integrators up to order four are constructed on the basis of the symmetry conditions. In order to show the remarkable performances of new symmetric methods in comparison with symmetric Runge–Kutta methods including the 2-stage Gauss method, two numerical experiments are carried out and the numerical results demonstrate the super numerical behavior.

The implementation of extrapolation with smoothing technique in solving stiff ordinary differential equations

PurposeExtrapolation is a process used to accelerate the convergence of a sequence of approximations to the true value. Different stepsizes are used to obtain approximate solutions, which are combined to increase the order of the approximation by eliminating leading error terms. The smoothing technique is also applied to suppress order reduction and to dampen the oscillatory component in the numerical solution when solving stiff problems. The extrapolation and smoothing technique can be applied in either active, passive or the combination of both active and passive modes. In this paper, the authors investigate the best strategy of implementing extrapolation and smoothing technique and use this strategy to solve stiff ordinary differential equations. Based on the experiment, the authors suggest using passive smoothing in order to reduce the computation time.Design/methodology/approachThe two-step smoothing is a composition of four steps of the symmetric method with different weights. It is used as the final two steps when combined with many steps of the symmetric method. The aim is to preserve symmetry and provide damping for stiff problem and to be more robust than the one-step smoothing. The two-step smoothing is L-stable. The new method is then applied with extrapolation process in passive and active modes to investigate the most efficient and accurate method of implementation.FindingsIn this paper, the authors constructed the two-step smoothing to be more robust than the one-step smoothing. The two-step smoothing is constructed to achieve as high order as possible and able to restore the classical order of particular method compared to the one-step active smoothing that is only able to achieve order-1 condition. The two-step smoothing for ITR is also superior in solving stiff case since it has the super-convergent order-4 behavior. In our experiments with extrapolation, it is proven that the two-step smoothing is more accurate and more efficient than the one-step smoothing, namely 1ASAX. It is also observed that the method with smoothing is comparable if not superior to the existing base method in certain cases. Based on the experiment, the authors would suggest using passive smoothing if the aim is to reduce computation time. It is of interest to conduct more experiment to validate the accuracy and efficiency of the smoothing formula with and without extrapolation.Originality/valueThe implementation of extrapolation on two-step symmetric Runge–Kutta method has not been tested on variety of other test problems yet. The two-step symmetrization is an extension of the one-step symmetrization and has not been constructed by other researchers yet. The method is constructed such that it preserves the asymptotic error expansion in even powers of stepsize, and when used with extrapolation the order might increase by 2 at a time. The method is also L-stable and eliminates the order reduction phenomenon when solving stiff ODEs. It is also of interest to observe other ways of implementing extrapolation using other sequences or with interpolation.

Extrapolation of Explicit DIMSIMs of High Order to Solve the Ordinary Differential Equations

The purpose of this research is to investigate the effciency of explicit diagonally implicit multi-stage integration methods with extrapolation. The author gave detailed explanation of explicit diagonally implicit multi-stage integration method and compared the base method with a technique known as extrapolation to improve the effciency. Extrapolation for symmetric Runge-Kutta method is proven to improve the accuracy since with extrapolation the solutions exhibit asymptotic error expansion, however for General linear methods, it is not known whether extrapolation can improve the effciency or not. Therefore this research focuses on the numerical experimental results of the explicit diagonally implicit multistage integration with and without extrapolation for solving some ordinary differential equations. The numerical results showed that the base method with extrapolation is more effcient than the method without extrapolation.

Delay-dependent stability of symmetric Runge–Kutta methods for second order delay differential equations with three parameters

The paper is concerned with the delay-dependent stability analysis of symmetric Runge–Kutta methods, which include the Gauss methods and the Lobatto IIIA, IIIB and IIIS methods, for the second order delay differential equations with three parameters. By using the root locus technique, the root locus curve is given and the numerical stability region of symmetric Runge–Kutta methods is obtained. It is proved that, under some conditions, the analytical stability region is contained in the numerical stability region. Numerical examples confirming the theoretical results are presented.

Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields

We apply a second-order symmetric Runge–Kutta method and a second-order symplectic Runge–Kutta method directly to the gyrocenter dynamics which can be expressed as a noncanonical Hamiltonian system. The numerical simulation results show the overwhelming superiorities of the two methods over a higher order nonsymmetric nonsymplectic Runge–Kutta method in long-term numerical accuracy and near energy conservation. Furthermore, they are much faster than the midpoint rule applied to the canonicalized system to reach given precision.

Symmetric general linear methods

The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge–Kutta methods, are applicable to general time-reversible differential equations, not just those derived from separable second-order problems. A definition of time-reversal symmetry is formulated for general linear methods, and criteria are found for the methods to be free of linear parasitism. It is shown that symmetric parasitism-free methods cannot be explicit, but such a method of order 4 is constructed with only one implicit stage. Several characterizations of symmetry are given, and connections are made with G-symplecticity. Symmetric methods are shown to be of even order, a suitable symmetric starting method is constructed and shown to be essentially unique. The underlying one-step method is shown to be time-symmetric.

Behaviour of the Extrapolated Implicit IMR and ITR With and Without Compensated Summation

Extrapolation involves taking a certain linear combination of the numerical solutions of a base method applied with different stepsizes to obtain greater accuracy. This linear combination is done so as to eliminate the leading error term. The technique of extrapolation in accelerating convergence has been successfully in numerical solution of ordinary differential equations. In this study, symmetric Runge-Kutta methods for solving linear and nonlinear stiff problem are considered. Symmetric methods admit asymptotic error expansion in even powers of the stepsize and are therefore of special interest because successive extrapolations can increase the order by two at time. Although extrapolation can give greater accuracy, due to the stepsize chosen, the numerical approximations are often destroy due to the accumulated round off errors. Therefore, it is important to control the rounding errors especially when applying extrapolation. One way to minimize round off errors is by applying compensated summation. In this paper, the numerical results are given for the symmetric Runge-Kutta methods Implicit Midpoint and Implicit Trapezoidal Rule applied with and without compensated summation. The result shows that symmetric methods with higher level extrapolation using compensated summation gives much smaller errors. On the other hand, symmetric methods without compensated summation when applied with extrapolation, the errors are affected badly by rounding errors

Active and passive symmetrization of Runge–Kutta Gauss methods

A symmetrizer for a symmetric Runge–Kutta method is designed to preserve the asymptotic error expansion in even powers of the stepsize and to provide damping for stiff initial value problems. In this paper we study symmetrizers for the Gauss methods with two and three stages and compare the implementation in passive and active modes. In particular, we perform a detailed analysis of the Prothero–Robinson problem which provides insight into the behaviour of symmetrizers in suppressing order reduction experienced by the symmetric methods. We present numerical results on the effects of passive and active symmetrization for some stiff linear and nonlinear problems. These effects have important implications for the development of extrapolation methods based on higher order symmetric methods for the numerical solution of stiff problems. Our results show that symmetrization in both modes improves accuracy and efficiency, and can restore the classical order of the Gauss methods for stiff linear problems. We compare the two modes of symmetrization for constant stepsize and present preliminary results in a variable stepsize setting.

Reversible methods of Runge-Kutta type for Index-2 DAEs

A new interpretation of Runge-Kutta methods for differential algebraic equations (DAEs) of index 2 is presented, where a step of the method is described in terms of a smooth map (smooth also with respect to the stepsize). This leads to a better understanding of the convergence behavior of Runge-Kutta methods that are not stiffly accurate. In particular, our new framework allows for the unified study of two order-improving techniques for symmetric Runge-Kutta methods (namely post-projection and symmetric projection) specially suited for solving reversible index-2 DAEs.

Symmetric Runge-Kutta methods and their stability

In this paper we find and study the class of symmetric methods among the Runge-Kutta formulae. It is shown that the explicit Runge-Kutta methods cannot be symmetric. We also define the conditions which coefficients in the implicit Runge-Kutta method should satisfy for it to be symmetric. Particular attention has been given to the study of stability properties in the symmetric Runge-Kutta formulae. It is proved that in some cases the notions of absolute and algebraic stability for the given class of numerical methods coincide. Besides, we find a restriction to the order of stable symmetric methods among the diagonally implicit Runge-Kutta formulae. Finally, we give full characteristics of all algebraically stable symmetric Runge-Kutta methods in terms of a transformed matrix of coefficients.

Symmetric Runge-Kutta methods and their stability

Symmetric Runge-Kutta methods and their stability

Generalized symmetric Runge-Kutta methods

In this paper the concept of symmetry for Runge-Kutta methods is generalized to include composite methods. The extrapolations of the usual compositions of a symmetric method ℛ of the form Open image in new window are shown not to beA-stable. However, this limitation can be overcome by considering composite methods of the form Open image in new window where Open image in new window represents a non-symmetric and possiblyL-stable method called a symmetrizer satisfying Open image in new window . While no longer symmetric, these composite methods yet satisfy Open image in new window and thus share with symmetric methods the important property of admitting asymptotic error expansions in even powers of 1/n. Composite methods that are constructed in this way and presented in this paper have implementation costs comparable to that for the symmetric method. They generalize those based on the implicit midpoint and trapezoidal rules used with the standard smoothing formulae and thus extend the class of methods for acceleration techniques of extrapolation and defect correction. A characterization ofL-stable symmetrizers for 2-stage symmetric methods is given and studied for a particular stiff model problem. The analysis suggests that certainL-stable symmetrizers can play an important role in suppressing order defect effects for stiff problems.

On symmetric Runge-Kutta methods of high order

The usual characterization of symmetry for Runge-Kutta methods is that given by Stetter. In this paper an equivalent characterization of symmetry based on theW-transformation of Hairer and Wanner is proposed. Using this characterization it is simple to show symmetry for some well-known classes of high order Runge-Kutta methods which are based on quadrature formulae. It can also be used to construct a one-parameter family of symmetric and algebraically stable Runge-Kutta methods based on Lobatto quadrature. Methods constructed in this way and presented in this paper extend the known class of implicit Runge-Kutta methods of high order.