The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrödinger Equation
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Alexander Zlotnik
and
Ilya Zlotnik
Abstract
We consider the Cauchy problem for the 1D generalized Schrödinger equation on the whole axis. To solve it, any order finite element in space and the Crank–Nicolson in time method with the discrete transparent boundary conditions (TBCs) has recently been constructed. Now we engage the global Richardson extrapolation in time to derive the high order method both with respect to space and time steps. To study its properties, we comment on its stability and give results of numerical experiments and enlarged practical error analysis for three typical examples. Unlike most common second order (in either space or time step) methods, the proposed method is able to provide high precision results in the uniform norm by using adequate computational costs. It works even in the case of discontinuous potentials and non-smooth solutions far beyond the scope of its standard theory. Comparing our results to the previous ones, we obtain much more accurate results using much less amount of both elements and time steps.
Funding source: The National Research University – Higher School of Economics' Academic Fund Program in 2014–2015
Award Identifier / Grant number: 14-01-0014
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 14-01-90009-Bel
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations
- Finite Element Analysis of an Exponentially Graded Mesh for Singularly Perturbed Problems
- A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem
- On the Theory of TE Waves Guided by a Lossy Three-Layer Structure with General Nonlinear Permittivity
- Analysis of Model Variance for Ensemble Based Turbulence Modeling
- On the Finite Volume Multigrid Method: Comparison of Intergrid Transfer Operators
- Approximation of Semilinear Fractional Cauchy Problem
- Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation
- On Conditioning of Constraints Arising from Variationally Consistent Discretization of Contact Problems and Duality Based Solvers
- The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrödinger Equation
Articles in the same Issue
- Frontmatter
- Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations
- Finite Element Analysis of an Exponentially Graded Mesh for Singularly Perturbed Problems
- A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem
- On the Theory of TE Waves Guided by a Lossy Three-Layer Structure with General Nonlinear Permittivity
- Analysis of Model Variance for Ensemble Based Turbulence Modeling
- On the Finite Volume Multigrid Method: Comparison of Intergrid Transfer Operators
- Approximation of Semilinear Fractional Cauchy Problem
- Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation
- On Conditioning of Constraints Arising from Variationally Consistent Discretization of Contact Problems and Duality Based Solvers
- The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrödinger Equation
