Integration Of Second-order Initial-value Problems Research Articles

A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

A new optimized symmetric 8-step semi-embedded predictor–corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems

In this work we introduce a new predictor-corrector (PC) pair form (SEPCM) for the numerical integration of second-order initial-value problems and a new optimized eight-step symmetric predictor-corrector method with minimal phase- lag and algebraic order ten is constructed. The new method is based on the multistep symmetric method of Quinlan-Tremaine (Astron. J. 100(5):1694-1700, 1990), with eight steps and eighth algebraic order and constructed to solve numerically the radial time-independent Schrodinger equation during the resonance problem with the use of the Woods-Saxon potential. It can also be used to integrate related IVPs with oscilla- tory solutions such as orbital problems. We compare the new method to some recently constructed optimized methods and other methods from the literature. We measure the efficiency of the methods and conclude that the new optimized method is the most efficient of all the compared methods and for all the problems solved.

A NEW EIGHT-STEP SYMMETRIC EMBEDDED PREDICTOR-CORRECTOR METHOD (EPCM) FOR ORBITAL PROBLEMS AND RELATED IVPs WITH OSCILLATORY SOLUTIONS

A new multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new eight-step symmetric embedded predictor-corrector method is constructed. The new PC method is based on the multistep symmetric method of Quinlan & Tremaine, with eight steps and eighth algebraic order, and is constructed to solve numerically the N-body problem. The new integrator has algebraic order 10 and it can be used to solve problems, for which the frequency is not known. We investigate the behavior of the new algorithm by integrating the five outer-planet problem and the two-body problem with various eccentricities. Regarding the five outer-planet problem, we calculate the error of the integrator in the solution, the Hamiltonian, and the phase after forward and backward integration over various intervals that are multiples of the period of Jupiter.

A NEW SYMMETRIC EIGHT-STEP PREDICTOR-CORRECTOR METHOD FOR THE NUMERICAL SOLUTION OF THE RADIAL SCHRÖDINGER EQUATION AND RELATED ORBITAL PROBLEMS

A new general multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new symmetric eight-step predictor-corrector method with minimal phase-lag and algebraic order ten is also constructed. The new method is based on the multistep symmetric method of Quinlan–Tremaine,1with eight steps and 8th algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.

Singly-implicit stabilized extended one-step methods for second-order initial-value problems with oscillating solutions

We introduce a new class of singly-implicit extended one-step methods for the numerical integration of second-order initial-value problems y″ = f( t, y), y( t 0) = η 0, y′( t 0) = η 1, with oscillating solutions. We first show that for third order, with two stages there exists a uniquely determined ‘almost’ P-stable method. We then investigate stability of the general class of fourth-order one-step methods. We first look for stabilized fourth-order methods with two stages, and show the interesting result that there exist families of two-stage fourth-order P-stable methods. We also obtain some families of three-stage fourth-order P-stable methods. The obtained methods are computationally tested on problems of practical interest.

An explicit four-step phase-fitted method for the numerical integration of second-order initial-value problems

An explicit four-step method with phase-lag of infinite order is developed for the numerical integration of second-order initial-value problems. Extensive numerical testing indicates that this new method can be generally more efficient than other four-step methods.