Optimizing a Hybrid Two-Step Method for the Numerical Solution of the Schrödinger Equation and Related Problems with Respect to Phase-Lag

We use a methodology of optimization of the efficiency of a hybrid two-step method for the numerical solution of the radial Schrödinger equation and related problems with periodic or oscillating solutions. More specifically, we study how the vanishing of the phase-lag and its derivatives optimizes the efficiency of the hybrid two-step method.

A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation

10.1007/s10910-015-0571-x ◽

2015 ◽

Vol 54 (2) ◽

pp. 442-465 ◽

Cited By ~ 63

Author(s):

Zhou Zhou ◽

T. E. Simos

Keyword(s):

Schrödinger Equation ◽

Numerical Solution ◽

Schrodinger Equation ◽

Phase Lag ◽

Step Method ◽

Two Stage ◽

A new family of three-stage two-step P-stable multiderivative methods with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation and IVPs with oscillating solutions

Numerical Algorithms ◽

10.1007/s11075-018-0497-z ◽

2018 ◽

Vol 80 (2) ◽

pp. 557-593 ◽

Cited By ~ 2

Author(s):

Ali Shokri ◽

Mohammad Mehdizadeh Khalsaraei ◽

Mortaza Tahmourasi ◽

Raquel Garcia-Rubio

Keyword(s):

Schrödinger Equation ◽

Numerical Solution ◽

Schrodinger Equation ◽

Phase Lag ◽

New Family ◽

Oscillating Solutions ◽

Radial Schrödinger Equation ◽

Multiderivative Methods

A new four-step P-stable Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2018.04.024 ◽

2019 ◽

Vol 354 ◽

pp. 569-586 ◽

Cited By ~ 3

Author(s):

Ali Shokri ◽

Jesús Vigo-Aguiar ◽

Mohammad Mehdizadeh Khalsaraei ◽

Raquel Garcia-Rubio

Keyword(s):

Schrödinger Equation ◽

Numerical Solution ◽

Schrodinger Equation ◽

Phase Lag ◽

A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schrödinger equation

10.1007/s10910-011-9897-1 ◽

2011 ◽

Vol 49 (10) ◽

pp. 2486-2518 ◽

Cited By ~ 43

Author(s):

T. E. Simos

Keyword(s):

Schrödinger Equation ◽

Numerical Solution ◽

Schrodinger Equation ◽

Phase Lag ◽

Step Method

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

10.1007/s10910-016-0600-4 ◽

2016 ◽

Vol 54 (4) ◽

pp. 1010-1040 ◽

Cited By ~ 17

Author(s):

Ibraheem Alolyan ◽

T. E. Simos

Keyword(s):

Schrödinger Equation ◽

Numerical Solution ◽

Schrodinger Equation ◽

Phase Lag ◽

A family of two stages tenth algebraic order symmetric six-step methods with vanished phase-lag and its first derivatives for the numerical solution of the radial Schrödinger equation and related problems

10.1007/s10910-016-0654-3 ◽

2016 ◽

Vol 54 (9) ◽

pp. 1835-1862 ◽

Cited By ~ 15

Author(s):

Ibraheem Alolyan ◽

T. E. Simos

Keyword(s):

Schrödinger Equation ◽

Numerical Solution ◽

Schrodinger Equation ◽

Phase Lag ◽

Radial Schrödinger Equation ◽

Algebraic Order ◽

Two Stages

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

International Journal of Modern Physics C ◽

10.1142/s0129183111016154 ◽

2011 ◽

Vol 22 (02) ◽

pp. 133-153 ◽

Cited By ~ 48

Author(s):

G. A. PANOPOULOS ◽

Z. A. ANASTASSI ◽

T. E. SIMOS

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

New Method ◽

Phase Lag ◽

Oscillating Solutions ◽

Orbital Problems ◽

Radial Schrödinger Equation ◽

Algebraic Order ◽

Predictor Corrector ◽

Minimal Phase

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,1 with 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.