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T. E. Simos - One of the best experts on this subject based on the ideXlab platform.
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an new high algebraic order efficient finite difference method for the solution of the schr odinger equation
Filomat, 2017Co-Authors: Ming Dong, T. E. SimosAbstract:The development of a new five-stages symmetric two-step method of fourteenth algebraic order with vanished phase--lag and its first, second, third and fourth derivatives is analyzed in this paper. More specifically : (1) we will present the development of the new method, (2) we will determine the Local Truncation Error (LTE) of the new proposed method, (3) we will analyze the Local Truncation Error based on the radial time independent Schr\"odinger equation, (4) we will study the stability and the interval of periodicity of the new proposed method based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase--lag analysis, (5) we will test the efficiency of the new obtained method based on its application on the coupled differential equations arising from the Schr\"odinger equation.
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A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation
Filomat, 2017Co-Authors: Ming Dong, T. E. SimosAbstract:The development of a new five-stages symmetric two-step method of fourteenth algebraic order with vanished phase-lag and its first, second, third and fourth derivatives is analyzed in this paper. More specifically: (1) we will present the development of the new method, (2) we will determine the Local Truncation Error (LTE) of the new proposed method, (3) we will analyze the Local Truncation Error based on the radial time independent Schrodinger equation, (4) we will study the stability and the interval of periodicity of the new proposed method based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, (5) we will test the efficiency of the new obtained method based on its application on the coupled differential equations arising from the Schrodinger equation.
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A family of three stages embedded explicit six–step methods with eliminated phase-lag and its derivatives for the numerical solution of second order problems
2017Co-Authors: T. E. SimosAbstract:In the present paper we investigate a family of three stages high algebraic order embedded explicit six–step methods with eliminated phase-lag and its derivatives for the numerical solution of second order periodic initial or boundary-value problems. The basis of the construction of the new proposed family of embedded methods is (1) the elimination phase–lag and (2) the elimination of the phase-lag’s derivatives.The produced methods with the above mentioned methodology are studied on: their Local Truncation Error, the asymptotic form of the Local Truncation Error which is produced applying them to the radial Schrodinger equation, the comparison of the asymptotic forms of the Local Truncation Errors which leads to conclusions on the effectiveness of each method of the family, the stability and the interval of periodicity of the developed methods of the new family of embedded pairs, the application the new obtained family of embedded methods to the numerical solution of several second order problems like th...
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An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems
Journal of Mathematical Chemistry, 2016Co-Authors: Ibraheem Alolyan, T. E. SimosAbstract:In this paper we develop a new implicit eighth algebraic order symmetric six-step method. For this method we request for the first time in the literature vanishing of the phase-lag and its first, second, third and fourth derivatives. The investigation of the new method consists of the following: the development of the method, i.e. the production of the coefficients of the method in order the phase-lag and the derivatives of the phase-lag to be vanished, the computation of the formula of the Local Truncation Error, the application of the new obtained method to a test problem (the radial Schrodinger equation) and the production of the asymptotic form of the Local Truncation for this test problem the comparison of the asymptotic forms of the Local Truncation Error of known similar methods with the asymptotic form Local Truncation Error of of the new developed method (comparative Local Truncation Error analysis) the stability investigation of the new produced method. This investigation is taken place by using a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the produced results i.e. by studying the interval of periodicity of the developed method. Finally we will apply the new developed method to the approximate solution of the resonance problem of the radial Schrodinger equation. The above mentioned application will help us on the study of the efficiency of the new obtained method. We will test the efficiency of the produced method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.
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An Efficient Numerical Method for the Solution of the Schrödinger Equation
Advances in Mathematical Physics, 2016Co-Authors: Licheng Zhang, T. E. SimosAbstract:The development of a new five-stage symmetric two-step fourteenth-algebraic order method with vanished phase-lag and its first, second, and third derivatives is presented in this paper for the first time in the literature. More specifically we will study the development of the new method, the determination of the Local Truncation Error (LTE) of the new method, the Local Truncation Error analysis which will be based on test equation which is the radial time independent Schrodinger equation, the stability and the interval of periodicity analysis of the new developed method which will be based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, and the efficiency of the new obtained method based on its application to the coupled Schrodinger equations.
Ibraheem Alolyan - One of the best experts on this subject based on the ideXlab platform.
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New four-stages symmetric six-step method with improved phase properties for second order problems with periodical and/or oscillating solutions
Journal of Mathematical Chemistry, 2018Co-Authors: Ibraheem Alolyan, T. E. SimosAbstract:In this paper, we build, for the first time in the literature, a new four-stages symmetric six-step finite difference pair with optimized properties. The method: 1. is a symmetric non-linear six-step method, 2. is of four stages 3. is of fourteenth algebraic order, 4. has eliminated the phase-lag, 5. has eliminated the first and second derivatives of the phase-lag. An analysis of the new proposed method is given in details in this paper. More specifically, we present: 1. the building of the new four-stages symmetric six-step method, 2. the computation of the Local Truncation Error of the new proposed method, 3. the comparative Local Truncation Error analysis of the new proposed method with other finite difference pairs of the same family. 4. the stability and the interval of periodicity analysis and 5. finally, the investigation and evaluation of the computational efficiency of the new proposed scheme for the approximate solution of the Schrödinger equation. The theoretical, computational and numerical results for the new proposed method show its effectiveness compared with other known or recently obtained finite difference pairs in the literature.
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An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems
Journal of Mathematical Chemistry, 2016Co-Authors: Ibraheem Alolyan, T. E. SimosAbstract:In this paper we develop a new implicit eighth algebraic order symmetric six-step method. For this method we request for the first time in the literature vanishing of the phase-lag and its first, second, third and fourth derivatives. The investigation of the new method consists of the following: the development of the method, i.e. the production of the coefficients of the method in order the phase-lag and the derivatives of the phase-lag to be vanished, the computation of the formula of the Local Truncation Error, the application of the new obtained method to a test problem (the radial Schrodinger equation) and the production of the asymptotic form of the Local Truncation for this test problem the comparison of the asymptotic forms of the Local Truncation Error of known similar methods with the asymptotic form Local Truncation Error of of the new developed method (comparative Local Truncation Error analysis) the stability investigation of the new produced method. This investigation is taken place by using a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the produced results i.e. by studying the interval of periodicity of the developed method. Finally we will apply the new developed method to the approximate solution of the resonance problem of the radial Schrodinger equation. The above mentioned application will help us on the study of the efficiency of the new obtained method. We will test the efficiency of the produced method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.
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A family of two-stages implicit six-step methods with vanished phase-lag and its derivatives
2016Co-Authors: Ibraheem Alolyan, T. E. SimosAbstract:A family of two-stages implicit symmetric six-step methods for the numerical integration of second order initial or boundary-value problems with periodical and / or oscillating solutions is studied in this paper. The development of the new family of methods is based on: the maximization of the algebraic order of the methods of the familythe vanishing phase-lag andthe vanished of the derivatives of the phase-lag.The study of the produced methods consists of the determination of the Local Truncation Error of the methodsthe Error analysis which is based on the application of the resulting methods to a test problem.the comparison of the produced asymptotic forms of the Local Truncation Error of the methods with the asymptotic forms of the other similar known methodsthe investigation of the stability for the obtained methods of the new family of methods.the application of the new developed family of methods to the radial time independent Schrodinger equation in order to show their efficiency.
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A high algebraic order predictor–corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems
Journal of Mathematical Chemistry, 2015Co-Authors: Ibraheem Alolyan, T. E. SimosAbstract:In this paper an eighth algebraic order predictor–corrector explicit four-step method is studied. The main scope of this paper is to study the consequences of (1) the vanishing of the phase-lag and its first, second, third and fourth derivatives and (2) the high algebraic order on the efficiency of the new developed method. A theoretical and computational study of the obtained method is also presented. More specifically, the theoretical study of the new predictor–corrector method consists of: The development of the new predictor–corrector method, i.e. the definition of the coefficients of the method in order its phase-lag and phase-lag’s first, second, third and fourth derivatives to be vanished The computation of the Local Truncation Error The comparative Local Truncation Error analysis The stability (interval of periodicity) analysis, using scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis. Finally, the computational study of the new predictor–corrector method consists of the application of the new produced predictor–corrector explicit four-step method to the numerical solution of the resonance problem of the radial time independent Schrödinger equation.
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efficient low computational cost hybrid explicit four step method with vanished phase lag and its first second third and fourth derivatives for the numerical integration of the schrodinger equation
Journal of Mathematical Chemistry, 2015Co-Authors: Ibraheem Alolyan, T. E. SimosAbstract:Based on an optimized explicit four-step method, a new hybrid high algebraic order four-step method is introduced in this paper. For this new hybrid method, we investigate the procedure of vanishing of the phase-lag and its first, second, third and fourth derivatives. More specifically, we investigate: (1) the construction of the new method, i.e. the computation of the coefficients of the method in order its phase-lag and first, second, third and fourth derivatives of the phase-lag to be eliminated, (2) the definition of the Local Truncation Error, (3) the analysis of the Local Truncation Error, (4) the stability (interval of periodicity) analysis (using scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis). Finally, we investigate computationally the new obtained method by applying it to the numerical solution of the resonance problem of the radial Schrodinger equation. The efficiency of the new developed method is tested comparing this method with well known methods of the literature but also using very recently developed methods.
A Idesman - One of the best experts on this subject based on the ideXlab platform.
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the use of the Local Truncation Error to improve arbitrary order finite elements for the linear wave and heat equations
Computer Methods in Applied Mechanics and Engineering, 2018Co-Authors: A IdesmanAbstract:Abstract The Local Truncation Error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high-order finite elements in the 1-D, 2-D and 3-D cases on uniform and non-uniform meshes. Several applications of the Local Truncation Error are considered in the paper. It is proven that for the 1-D wave equation with a piece-wise constant wave velocity, the Local Truncation Error is zero if the linear finite elements with the element size proportional to the wave velocity, the lumped mass matrix and the central-difference method with the time increments equal to the stability limit are used. It is shown in the 1-D and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the Local Truncation Error and yields the maximum possible order of accuracy. The minimization of the order of the Local Truncation Error allows us to develop the linear finite elements and the isogeometric high-order elements with improved accuracy; i.e., accuracy is improved from order 2 p (the conventional elements) to order 4 p (the new elements) where p is the order of the polynomial approximations. New high-order boundary conditions are developed in order to keep a high-order accuracy of the developed technique. The new elements can be equally applied to linear wave propagation and heat transfer problems. It is also shown that non-uniform meshes may lead to inaccurate results due to the increase in the Local Truncation Error. The difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach. The presented numerical examples are in good agreement with the theoretical results. The approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches.
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the use of the Local Truncation Error for the increase in accuracy of the linear finite elements for heat transfer problems
Computer Methods in Applied Mechanics and Engineering, 2017Co-Authors: A Idesman, B DeyAbstract:Abstract A new approach for the increase in the order of accuracy of the linear finite elements used for the time dependent heat equation and for the time independent Laplace equation has been suggested. It is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the Local Truncation Error. By a simple modification of the coefficients of the elemental mass and stiffness matrices, the accuracy of the linear finite elements is improved by two orders for the heat equation and by four orders for the Laplace equation. Despite the significant increase in accuracy, the computational costs of the new technique are the same as those for the conventional linear finite elements on a given mesh. 2- D and 3- D numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new linear finite elements are much more accurate than the conventional linear and quadratic finite elements at the same numbers of degrees of freedom.
Xu Ying-xiang - One of the best experts on this subject based on the ideXlab platform.
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Compactly supported spline wavelet method of initial value problem
Journal of Northwest Normal University, 2008Co-Authors: Xu Ying-xiangAbstract:Using cubic compactly supported spline wavelet interpolation function,the numerical solution of initial value problem of the first order ordinary differential equations is discussed,and an inexplicit approximation solving formula is given.Its Local Truncation Error is O(5),but the global Truncation Error is O(4) and it is stable.On this bases,an explicit revise formula for the problem is obtained,and the Local Truncation Error is discussed.A numerical example is given at last.
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Compactly Spline Wavelet Method of a Class of System of ODEs
Journal of Ningbo University, 2007Co-Authors: Xu Ying-xiangAbstract:Using cubic compactly spline wavelet interpolation function,the numerical solution of initial value problem of a class of ODEs system is discussed,and an inexplicit approximation solving formula is given.Its Local Truncation Error is O(5),and based on which an explicit revise formula for the problem is obtained and the Local Truncation Error is O(4).
Giovanni Lapenta - One of the best experts on this subject based on the ideXlab platform.
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variational grid adaptation based on the minimization of Local Truncation Error time independent problems
Journal of Computational Physics, 2004Co-Authors: Giovanni LapentaAbstract:A new approach to grid adaptation is presented. The method is based on two established foundations. First, the method is based upon variational grid adaptation, retaining all the well-known properties of robustness and regularity. Second, the adaptation method presented here is based on a general definition of the Error detector obtained from the moving finite element (MFE) method. The Error detector is general, applicable to any given problem, and does not require any a priori knowledge of the solution or of the physical behaviour of the system under investigation. The primary theoretical contribution of the present work is in establishing a link between various adaptation methods previously regarded as different and unrelated. We show that they all derive from the same approach and are all equivalent in the sense that the same grid is generated by all of them for the same problem, once the monitor functions are chosen according to our approach. The primary practical contribution of the present work is in prescribing a rigorous monitor function for previously published adaptation strategies. The choice proposed here is shown to outperform previous heuristic choices. The method is tested in a series of elliptic problems, where the adaptation strategy presented here can improve the accuracy by orders of magnitude.
