Zhongcheng Wang

P-stable linear symmetric multistep methods for periodic initial-value problems

In this paper we present a new kind of P-stable multistep methods for periodic initial-value problems. From the numerical results obtained by the new method to well-known periodic problems, show the superior efficiency, accuracy, stability of the method presented in this paper.

An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial-value problems

In this paper we present an improved P-stable trigonometrically fitted Obrechkoff method with phase-lag (frequency distortion) infinity. Compared with the previous P-stable trigonometrically fitted Obrechkoff method developed by Simos, our new method is simpler in structure and more stable in computation. We have also improved the accuracy of the first-order derivative formula. From the numerical illustration presented, we can show that the new method is much more accurate than the previous methods.

Importance of the first-order derivative formula in the Obrechkoff method

Abstract In this paper we present a delicately designed numerical experiment to explore the relationship between the accuracy of the first-order derivative (FOD) formula and the one of the main structure in an Obrechkoff method. We choose three two-step P-stable Obrechkoff methods as the main structure, which are available from the previous published literature, their local truncation error (LTE( h )) ranging from O ( h 8 y n ( 8 ) ) to O ( h 12 y n ( 12 ) ) , and six FOD formulas, of which the former five ones have the similar structures and the sixth is the ‘exact’ value of the FOD, their LTE( h ) arranged from O ( h 4 y n ( 5 ) ) to O ( h 13 y n ( 14 ) ) (we will use O ( y n ( m ) ) to represent the order of a LTE( h )), as the main ingredients for our numerical experiment. We survey the numerical results by integrating the Duffing equation without damping and compare them with the ‘exact’ solution, and find out how its numerical accuracy is affected by a FOD formula. The experiment shows that a high accurate FOD formula can greatly improve the numerical accuracy of an Obrechkoff method for a given main structure, and the error in the numerical solution decreases with the order of the LTE( h ) of a FOD formula, only when the order of LTE( h ) of the FOD formula is equal to or higher than the one of the main structure, the accuracy of the Obrechkoff method is no longer affected by the approximation of the FOD formula.

A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped Duffing's equation

Abstract Based on the idea of the previous Obrechkoffs two-step method, a new kind of four-step numerical method with free parameters is developed for the second order initial-value problems with oscillation solutions. By using high-order derivatives and apropos first-order derivative formula, the new method has greatly improved the accuracy of the numerical solution. Although this is a multistep method, it still has a remarkably wide interval of periodicity, H 0 2 ∼ 16.33 . The numerical test to the well known problem, the nonlinear undamped Duffings equation forced by a harmonic function, shows that the new method gives the solution with four to five orders higher than those by the previous Obrechkoffs two-step method. The ultimate accuracy of the new method can reach about 5 × 10 −13 , which is much better than the one the previous method could. Furthermore, the new method shows the great superiority in efficiency due to a reasonable arrangement of the structure. To finish the same computational task, the new method can take only about 20% CPU time consumed by the previous method. By using the new method, one can find a better ‘exact’ solution to this problem, reducing the error tolerance of the one widely used method ( 10 −11 ) , to below 10−14.

A trigonometrically-fitted one-step method with multi-derivative for the numerical solution to the one-dimensional Schrödinger equation

Abstract In this paper we present a new multi-derivative or Obrechkoff one-step method for the numerical solution to an one-dimensional Schrodinger equation. By using trigonometrically-fitting method (TFM), we overcome the traditional Obrechkoff one-step method (or called as the non-TFM) for its poor-accuracy in the resonant state. In order to demonstrate the excellent performance for the resonant state, we consider only the simplest TFM, of which the local truncation error (LTE) is of O ( h 7 ) , a little higher than the one of the traditional Numerov method of O ( h 6 ) , and only the first- and second-order derivatives of the potential function are needed. In the new method, in order to solve two unknowns, wave function and its first-order derivative, we use a pair of two symmetrically linear-independent one-step difference equations. By applying it to the well-known Woods–Saxons potential problem, we find that the TFM can surpass the non-TFM by five orders for the highest resonant state, and surpass Numerov method by eight orders. On the other hand, because of the small error constant, the accuracy improvement to the ground state is also remarkable, and the numerical result obtained by TFM can be four to five orders higher than the one by Numerov method.

A new kind of high-efficient and high-accurate P-stable Obrechkoff three-step method for periodic initial-value problems

Abstract In order to improve the efficiency and accuracy of the previous Obrechkoff method, in this paper we put forward a new kind of P-stable three-step Obrechkoff method of O ( h 10 ) for periodic initial-value problems. By using a new structure and an embedded high accurate first-order derivative formula, we can avoid time-consuming iterative calculation to obtain the high-order derivatives. By taking advantage of new trigonometrically-fitting scheme we can make both the main structure and the first-order derivative formula to be P-stable. We apply our new method to three periodic problems and compare it with the previous three Obrechkoff methods. Numerical results demonstrate that our new method is superior over the previous ones in accuracy, efficiency and stability.

A new effective algorithm for the resonant state of a Schrödinger equation

Abstract In this paper we present a new effective algorithm for the Schrodinger equation. This new method differs from the original Numerov method only in one simple coefficient, by which we can extend the interval of periodicity from 6 to infinity and obtain an embedded correct factor to improve the accuracy. We compare the new method with the original Numerov method by the well-known problem of Woods–Saxon potential. The numerical results show that the new method has great advantage in accuracy over the original. Particularly for the resonant state, the accuracy is improved with four orders overall, and even six to seven orders for the highest oscillatory solution. Surely, this method will replace the original Numerov method and be widely used in various area.

Arbitrarily precise numerical solutions of the one-dimensional Schrödinger equation

Abstract In this paper, how to overcome the barrier for a finite difference method to obtain the numerical solutions of a one-dimensional Schrodinger equation defined on the infinite integration interval accurate than the computer precision is discussed. Five numerical examples of solutions with the error less than 10 −50 and 10 −30 for the bound and resonant state, respectively, obtained by the Obrechkoff one-step method implemented in the multi precision mode, which include the harmonic oscillator, the Poschl–Teller potential, the Morse potential and the Woods–Saxon potential, demonstrate that the finite difference method can yield the eigenvalues of a complex potential with an arbitrarily desired precision within a reasonable efficiency.

Trigonometrically-fitted method for a periodic initial value problem with two frequencies

Abstract A second-order differential equation whose solution is periodic with two frequencies has important applications in many scientific fields. Nevertheless, it may exhibit ‘periodic stiffness’ for most of the available linear multi-step methods. The phenomena are similar to the popular Stomer–Cowell class of linear multi-step methods for one-frequency problems. According to the stability theory laid down by Lambert, ‘periodic stiffness’ appears in a two-frequency problem because the production of the step-length and the bigger angular frequency lies outside the interval of periodicity. On the other hand, for a two-frequency problem, even with a small step-length, the error in the numerical solution afforded by a P-stable trigonometrically-fitted method with one frequency would be too large for practical applications. In this paper we demonstrate that the interval of periodicity and the local truncation error of a linear multi-step method for a two-frequency problem can be greatly improved by a new trigonometric-fitting technique. A trigonometrically-fitted Numerov method with two frequencies is proposed and has been verified to be P-stable with vanishing local truncation error for a two-frequency test problem. Numerical results demonstrated that the proposed trigonometrically-fitted Numerov method with two frequencies has significant advantages over other types of Numerov methods for solving the ‘periodic stiffness’ problem.

A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach

Abstract According to Mickens [R.E. Mickens, Comments on a Generalized Galerkins method for non-linear oscillators, J. Sound Vib. 118 (1987) 563], the general HB (harmonic balance) method is an approximation to the convergent Fourier series representation of the periodic solution of a nonlinear oscillator and not an approximation to an expansion in terms of a small parameter. Consequently, for a nonlinear undamped Duffing equation with a driving force B cos ( ω x ) , to find a periodic solution when the fundamental frequency is identical to ω , the corresponding Fourier series can be written as y ˜ ( x ) = ∑ n = 1 m a n cos [ ( 2 n − 1 ) ω x ] . How to calculate the coefficients of the Fourier series efficiently with a computer program is still an open problem. For HB method, by substituting approximation y ˜ ( x ) into force equation, expanding the resulting expression into a trigonometric series, then letting the coefficients of the resulting lowest-order harmonic be zero, one can obtain approximate coefficients of approximation y ˜ ( x ) [R.E. Mickens, Comments on a Generalized Galerkins method for non-linear oscillators, J. Sound Vib. 118 (1987) 563]. But for nonlinear differential equations such as Duffing equation, it is very difficult to construct higher-order analytical approximations, because the HB method requires solving a set of algebraic equations for a large number of unknowns with very complex nonlinearities. To overcome the difficulty, forty years ago, Urabe derived a computational method for Duffing equation based on Galerkin procedure [M. Urabe, A. Reiter, Numerical computation of nonlinear forced oscillations by Galerkins procedure, J. Math. Anal. Appl. 14 (1966) 107–140]. Dooren obtained an approximate solution of the Duffing oscillator with a special set of parameters by using Urabes method [R. van Dooren, Stabilization of Cowells classic finite difference method for numerical integration, J. Comput. Phys. 16 (1974) 186–192]. In this paper, in the frame of the general HB method, we present a new iteration algorithm to calculate the coefficients of the Fourier series. By using this new method, the iteration procedure starts with a ( x ) cos ( ω x ) + b ( x ) sin ( ω x ) , and the accuracy may be improved gradually by determining new coefficients a 1 , a 2 , … will be produced automatically in an one-by-one manner. In all the stage of calculation, we need only to solve a cubic equation. Using this new algorithm, we develop a Mathematica program, which demonstrates following main advantages over the previous HB method: (1) it avoids solving a set of associate nonlinear equations; (2) it is easier to be implemented into a computer program, and produces a highly accurate solution with analytical expression efficiently. It is interesting to find that, generally, for a given set of parameters, a nonlinear Duffing equation can have three independent oscillation modes. For some sets of the parameters, it can have two modes with complex displacement and one with real displacement. But in some cases, it can have three modes, all of them having real displacement. Therefore, we can divide the parameters into two classes, according to the solution property: there is only one mode with real displacement and there are three modes with real displacement. This program should be useful to study the dynamically periodic behavior of a Duffing oscillator and can provide an approximate analytical solution with high-accuracy for testing the error behavior of newly developed numerical methods with a wide range of parameters. Program summary Title of program: AnalyDuffing.nb Catalogue identifier: ADWR_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWR_v1_0 Program obtainable from: CPC Program Library, Queens University of Belfast, N. Ireland Licensing provisions: none Computer for which the program is designed and others on which it has been tested: the program has been designed for a microcomputer and been tested on the microcomputer. Computers: IBM PC Installations: the address(es) of your computer(s) Operating systems under which the program has been tested: Windows XP Programming language used: Software Mathematica 4.2, 5.0 and 5.1 No. of lines in distributed program, including test data, etc.: 23 663 No. of bytes in distributed program, including test data, etc.: 152 321 Distribution format: tar.gz Memory required to execute with typical data: 51 712 Bytes No. of bits in a word: No. of processors used: 1 Has the code been vectorized?: no Peripherals used: no Program Library subprograms used: no Nature of physical problem: To find an approximate solution with analytical expressions for the undamped nonlinear Duffing equation with periodic driving force when the fundamental frequency is identical to the driving force. Method of solution: In the frame of the general HB method, by using a new iteration algorithm to calculate the coefficients of the Fourier series, we can obtain an approximate analytical solution with high-accuracy efficiently. Restrictions on the complexity of the problem: For problems, which have a large driving frequency, the convergence may be a little slow, because more iterative times are needed. Typical running time: several seconds Unusual features of the program: For an undamped Duffing equation, it can provide all the solutions or the oscillation modes with real displacement for any interesting parameters, for the required accuracy, efficiently. The program can be used to study the dynamically periodic behavior of a nonlinear oscillator, and can provide a high-accurate approximate analytical solution for developing high-accurate numerical method.