Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity

Abstract

We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schrödinger equation (NLSE) with semi-smooth nonlinearity f ( ρ ) = ρ σ f(\rho ) = \rho ^\sigma , where ρ = | ψ | 2 \rho =|\psi |^2 is the density with ψ \psi the wave function and σ > 0 \sigma >0 is the exponent of the semi-smooth nonlinearity. Under the assumption of H 2 H^2 -solution of the NLSE, we prove error bounds at O ( τ 1 2 + σ + h 1 + 2 σ ) O(\tau ^{\frac {1}{2}+\sigma } + h^{1+2\sigma }) and O ( τ + h 2 ) O(\tau + h^{2}) in L 2 L^2 -norm for 0 > σ ≤ 1 2 0>\sigma \leq \frac {1}{2} and σ ≥ 1 2 \sigma \geq \frac {1}{2} , respectively, and an error bound at O ( τ 1 2 + h ) O(\tau ^\frac {1}{2} + h) in H 1 H^1 -norm for σ ≥ 1 2 \sigma \geq \frac {1}{2} , where h h and τ \tau are the mesh size and time step size, respectively. In addition, when 1 2 > σ > 1 \frac {1}{2}>\sigma >1 and under the assumption of H 3 H^3 -solution of the NLSE, we show an error bound at O ( τ σ + h 2 σ ) O(\tau ^{\sigma } + h^{2\sigma }) in H 1 H^1 -norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional L 2 L^2 -stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of 0 > σ ≤ 1 2 0 > \sigma \leq \frac {1}{2} , and to establish an l ∞ l^\infty -conditional H 1 H^1 -stability to obtain the l ∞ l^\infty -bound of the numerical solution by using the mathematical induction and the error estimates for the case of σ ≥ 1 2 \sigma \ge \frac {1}{2} ; and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.

An Extended Fourier Pseudospectral Method for the Gross-Pitaevskii Equation with Low Regularity Potential

“…Let ψ n (•) be the numerical approximations of ψ(•, t n ) for n ≥ 0. Then the first-order Lie-Trotter time-splitting Fourier spectral (LTFS) method [11] reads…”

“…, d being some given real-valued constants. For the GPE (1.1) with sufficiently smooth potential, many accurate and efficient temporal discretizations have been proposed and analyzed in last two decades, including the finite difference time domain (FDTD) method [1,[5][6][7], the exponential wave integrator (EWI) [8,17,25], the time-splitting method [5,6,9,10,12,14,19,28,29], and the low regularity integrator (LRI) [2,4,16,27,[30][31][32]34]. Generally, these temporal discretizations are followed by a spatial discretization, such as finite difference methods, finite element methods or Fourier spectral/pseudospectral methods, to obtain a full discretization for the GPE (1.1).…”

“…Recently, much attention has been paid to the error analysis of those methods for tt the GPE with low regularity potential and/or the NLSE with low regularity nonlinearity. For details, we refer to [24] for the FDTD method, [13] for the EWI, [11,12,41] for the time-splitting method, and [2][3][4] for the LRI. Among these methods, under the assumption of L ∞ -potential, for a general H 2 -solution (which is the best situation that one can expect on the regularity of the exact solution due to the low regularity of potential), the optimal L 2 -norm error bound -first order in time and second order in space -can only be proved for the EWI and the time-splitting method with the Fourier spectral (FS) method for spatial discretization [11,13].…”

“…For details, we refer to [24] for the FDTD method, [13] for the EWI, [11,12,41] for the time-splitting method, and [2][3][4] for the LRI. Among these methods, under the assumption of L ∞ -potential, for a general H 2 -solution (which is the best situation that one can expect on the regularity of the exact solution due to the low regularity of potential), the optimal L 2 -norm error bound -first order in time and second order in space -can only be proved for the EWI and the time-splitting method with the Fourier spectral (FS) method for spatial discretization [11,13]. According to both theoretical and numerical results in [11,13], it is essential to use the FS method rather than the Fourier pseudospectral (FP) method to discretize the EWI and time-splitting methods in space in order to get optimal spatial convergence.…”

“…Among these methods, under the assumption of L ∞ -potential, for a general H 2 -solution (which is the best situation that one can expect on the regularity of the exact solution due to the low regularity of potential), the optimal L 2 -norm error bound -first order in time and second order in space -can only be proved for the EWI and the time-splitting method with the Fourier spectral (FS) method for spatial discretization [11,13]. According to both theoretical and numerical results in [11,13], it is essential to use the FS method rather than the Fourier pseudospectral (FP) method to discretize the EWI and time-splitting methods in space in order to get optimal spatial convergence. However, the FS method cannot be efficiently implemented in practice due to the difficulty in the exact evaluation of the Fourier integrals, and the usual strategy of using quadrature rules is extremely time-consuming to obtain the required accuracy due to the low regularity of potential.…”