Geometric Numerical Integration and Applications



 

  Coefficients of several symplectic splitting methods for integrating in time the Schrödinger equation


In the paper



we have designed several splitting methods for integrating in time the (previously discretized in space) Schrödinger equation. 
Their relevant parameters of the new methods are collected in Table 1 of the paper, which we reproduce here for the interested reader. In addition, we provide the coefficients of all the methods collected in the paper.
m
 r
 θ′
 y_∗/m
 ∑(|a_j| + |b_j|)
 μ_r(θ′m)
 ν_r(θ′m)
10
 6
 1
 1.1617
 4.022
 0.0009341
 0.0372
 Coefficients
20
 16
 1
 1.0456 3.0553 0.000611028 0.0258433 Coefficients
30
 24
 1
 1.0246 3.19658 0.0000841871 0.0373544 Coefficients
30
 6
 1.4
 1.41876
 3.0921
 0.0000518519
 0.0131295
 Coefficients
30
 0
 1
 1.1411
 3.04948
 2.91902 ⋅ 10^(-13)
 2.28673 ⋅ 10^(-9)
 Coefficients
30
 0
 0.75
 1.027
 3.44381
 1.2545 ⋅ 10^(-17) 5.96706 ⋅ 10^(-14) Coefficients
30
 0
 0.5
 0.937874
        3.84442 7.96031 ⋅ 10^(-24) 6.66693 ⋅ 10^(-18) Coefficients
40
 0
 1
   1.15953 3.21986 1.06301 ⋅ 10^(-15) 1.07587 ⋅ 10^(-12) Coefficients



a(i)  denote the coefficients a_i  in the composition, whereas  b(i) refer to the coefficients b_i.