Time staggering for wave equations revisited

Staggering in numerical methods for wave equations generally enhances accuracy and stability. This note is about time staggering. We assess a fourth-order, explicit, time-staggered method, while focussing on a class of second-order wave equations. Alternative explicit integration methods for this class belong to the Runge-Kutta-Nyström (RKN) family and we have selected three explicit RKN methods for our assessment of the time-staggered method. Compared to these three explicit RKN methods, the time-staggered method possesses a substantially longer stability interval. Our aim is to examine whether this advantage can be expected to borne out in actual computation

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations

Fifth order two-stage explicit Runge-Kutta-Nystrom method for the direct integration of second order ordinary differential equations

In this paper a direct integration of second-order Ordinary Differential Equations (ODEs) of the form using the Explicit Runge-Kutta-Nystrom method with higher derivatives is presented. Various numerical schemes are derived and tested on standard problems. The higher-order explicit Runge-Kutta-Nystrom (HERKN) method given in this paper is compared with the conventional Explicit Runge Kutta (ERK) schemes. Due to the limitation of ERK schemes in handling stiff problems, the extension to higher order derivative is considered. The results obtained show an improvement on ERK schemes

Stability of Runge–Kutta–Nyström methods

AbstractIn this paper, a general and detailed study of linear stability of Runge–Kutta–Nyström (RKN) methods is given. In the case that arbitrarily stiff problems are integrated, we establish a condition that RKN methods must satisfy so that a uniform bound for stability can be achieved. This condition is not satisfied by any method in the literature. Therefore, a stable method is constructed and some numerical comparisons are made

Parallel Störmer-Cowell methods for high-precision orbit computations

Many orbit problems in celestial mechanics are described by (nonstiff) initial-value problems (IVPs) for second-order ordinary differential equations of the form y' = {bf f (y). The most successful integration methods are based on high-order Runge-Kutta-Nyström formulas. However, these methods were designed for sequential is paper, we consider high-order parallel methods that are not based on Runge-Kutta-Nyström formulas, but which fit into the class of general linear methods. In each step, these methods compute blocks of k approximate solution values (or stage values) at k different points using the whole previous block of solution values. The k stage values can be computed in parallel, so that on a k-processor computer system such methods effectively perform as a one-value method. The block methods considered in this paper are such that each equation defining a stage value resembles a linear multistep equation of the familiar Störmer-Cowell type. For k = 4 and k = 5 we constructed explicit PSC methods with stage order q = k and step point order p = k+1 and implicit PSC methods with q = k+1 and p = k+2. For k = 6 we can construct explicit PSC methods with q = k and p = k+2 and implicit PSC methods with q = k+1 and p = k+3. It turns out that for k = 5 the abscissae of the stage values can be chosen such that only k-1 stage values in each block have to be computed, so that the number of computational stages, and hence the number of processors and the number of starting values needed, reduces to k* = k-1. The numerical examples reported in this paper show that the effective number of righthand side evaluations required by a variable stepsize implementation of the 10th-order PSC method is 4 up to 30 times less than required by the Runge-Kutta-Nyström code DOPRIN (which is considered as one of the most efficient sequential codes for second-order ODEs). Furthermore, a comparison with the 12th-order parallel code PIRKN reveals that the PSC code is, in spite of its lower order, at least equally efficient, and in most cases more efficient than PIRKN

Third derivative modification of k-step block Falkner methods for the numerical solution of second order initial-value problems

[EN]This paper is devoted to the development and analysis of a modified family of Falkner- type methods for solving differential systems of second-order initial-value problems. The approaches of collocation and interpolation are adopted to derive the new methods. These modified methods are implemented in block form to obtain the numerical solutions to the considered problems. The study of the properties of the proposed block Falkner-type methods reveals that they are consistent and zero-stable, and thus, convergent. From the stability analysis, it could be seen that the proposed Falkner methods have non-empty sta- bility regions for k = 2 , 3 , 4 . Some numerical test are presented to illustrate the efficiency of the proposed family

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

AbstractMany simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge–Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger 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. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic orde