A new variable-step method for the numerical integration of special second-order initial value problems and their application to the one- dimensional Schrödinger equation

DOI10.1016/0893-9659(93)90037-NzbMath0772.65048MaRDI QIDQ2367984

Theodore E. Simos

Publication date: 19 August 1993

Published in: Applied Mathematics Letters (Search for Journal in Brave)

zbMATH Keywords

finite difference methodssecond-order initial value problemsone- dimensional Schrödinger equationvariable-step methodphase-shift problem

Mathematics Subject Classification ID

Nonlinear ordinary differential equations and systems (34A34) Numerical methods for initial value problems involving ordinary differential equations (65L05) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Finite difference and finite volume methods for ordinary differential equations (65L12) Mesh generation, refinement, and adaptive methods for ordinary differential equations (65L50)

Related Items (10)

A new two-step hybrid method for the numerical solution of the Schrödinger equation ⋮ VSVO multistep formulae adapted to perturbed second-order differential equations ⋮ High algebraic order methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation ⋮ Mulitstep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrödinger equation ⋮ A hybrid method with phase-lag and derivatives equal to zero for the numerical integration of the Schrödinger equation ⋮ Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation ⋮ A new symmetric linear eight-step method with fifth trigonometric order for the efficient integration of the Schrödinger equation ⋮ A family of eight-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation ⋮ High order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation ⋮ A variable mesh C-SPLAGE method of accuracy \(O(k^2h_l^{-1}+kh_l+h_l^3)\) for 1D nonlinear parabolic equations

Cites Work

• A Numerov-type method for the numerical solution of the radial Schrödinger equation
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• A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: Explicit method
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• An explicit almost \(P\)-stable two-step method with phase-lag of order infinity for the numerical integration of second-order periodic initial- value problems
• An explicit sixth-order method with phase-lag of order eight for \(y=f(t,y)\)
• Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrödinger equation
• Explicit Runge–Kutta (–Nyström) Methods with Reduced Phase Errors for Computing Oscillating Solutions
• Numerical Methods for y″ =f(x, y) via Rational Approximations for the Cosine
• A one-step method for direct integration of structural dynamic equations
• A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problem
• Comparing Error Estimators for Runge-Kutta Methods

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