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A space-time DG method for the Schrödinger equation with variable potential

Sergio Gómez, Andrea Moiola

Abstract

We present a space-time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal $h$-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials, or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.

A space-time DG method for the Schrödinger equation with variable potential

Abstract

We present a space-time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal -convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials, or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.
Paper Structure (26 sections, 9 theorems, 59 equations, 13 figures, 4 tables)

This paper contains 26 sections, 9 theorems, 59 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Ultra-weak discontinuous Galerkin formulation
  3. Space--time mesh and DG notation
  4. Variational formulation of the DG method
  5. Well-posedness, stability and quasi-optimality of the DG method
  6. Discrete spaces and error estimates
  7. Multi-index notation and preliminary results
  8. Full polynomial space
  9. Quasi-Trefftz spaces
  10. Basis functions and dimension
  11. Numerical experiments
  12. (1+1)-dimensional test cases
  13. $h$-convergence
  14. Harmonic oscillator potential ($V(x) = \frac{\omega^2 x^2}{2}$)
  15. Reflectionless potential ($V(x) = -a^2\mathrm{sech}^2(ax)$)
  16. ...and 11 more sections

Key Result

Proposition 1

For all $w \in {{\mathbf{V}}}({{\mathcal{T}}_h})$ the following identity holds

Figures (13)

  • Figure 1: A representation of the relations defining the coefficients of $b_J$ for the (1+1)-dimensional case. The colored dots in the $(j_x,j_t)$ plane represent the coefficients $C_{j_x\,j_t}$. Each shape connects three dots located at the points $(j_x,j_t+1)$, $(j_x,j_t)$ and $(j_x+2,j_t)$: this shape represents one of the equations \ref{['EQN::TAYLOR-COEFFICIENTS-RELATION']} which, given $C_{j_x(j_t+1)}$ and $C_{j_x\,j_t}$, allows to compute $C_{(j_x+2)j_t}$. If the $2p+1$ values with $j_x\in\{0,1\}$ (corresponding to the blue nodes in the shaded region) are given, then these relations uniquely determine all the other coefficients, which can be computed sequentially using the relations \ref{['EQN::TAYLOR-COEFFICIENTS-RELATION']} by proceeding left to right in the diagram. In the figure $p=7$, the number of nodes is $r_{2,p}=36$, the number of nodes in the shaded region is $n_{2,p}=15$, the number of relations is $r_{2,p}-n_{2,p}=21$.
  • Figure 2: A representation of the relations defining the coefficients of $b_J$ for the (2+1)-dimensional case. The colored dots in position ${\boldsymbol j}=(j_x,j_y,j_t)$, $|{\boldsymbol j}|\le p$, correspond to the coefficients $C_{j_x\,j_y\,j_t}$ (here $p=5$ and $r_p=56$). Each white circle is connected by the segments to four nodes and represents one of the equations in \ref{['EQN::TAYLOR-COEFFICIENTS-RELATION']}: given $C_{j_x\,j_y\,j_t}$, $C_{j_x\,j_y(j_t+1)}$ and $C_{j_x(j_y+2)j_t}$, it allows to compute $C_{(j_x+2)j_y\,j_t}$ (the leftmost of the four nodes connected to a given white circle) using \ref{['EQN::TAYLOR-COEFFICIENTS-RELATION']}. The red dot exemplifies one of these relations, for ${\boldsymbol j}=(0,1,2)$. Given the $(p+1)^2$ coefficients with $j_x\in\{0,1\}$ (the blue dots), all other coefficients are uniquely determined.
  • Figure 3: Time-evolution of the energy error for the quantum harmonic oscillator problem with potential $\left(V(x) = 50 x^2\right)$ and exact solution $\psi_2$ in \ref{['EQN::EXACT-SOLUTION-HARMONIC']}.
  • Figure 4: $h$-convergence for the $(1+1)$ quantum harmonic oscillator problem with potential $\left(V(x) = 50 x^2\right)$ and exact solution $\psi_2$ in \ref{['EQN::EXACT-SOLUTION-HARMONIC']}. Convergence with respect to the mesh size $h$ (top panels) and the total number of degrees of freedom (bottom panels).
  • Figure 5: $h$-convergence for the $(1+1)$ problem with potential $V(x) = -\text{sech}^2(x)$ and exact solution \ref{['EQN::EXACT-SOLUTION-SECH']}.
  • ...and 8 more figures

Theorems & Definitions (25)

  • Remark 1: Implicit time-stepping through time-slabs
  • Remark 2: Self-adjointness and volume penalty term
  • Remark 3: Time-dependent potentials
  • Proposition 1: Coercivity
  • proof
  • Proposition 2: Continuity
  • proof
  • Theorem 1: Quasi-optimality
  • proof
  • Proposition 3
  • ...and 15 more