Abstract
We compare adaptive time integrators for the numerical solution of linear Schrödinger equations where the Hamiltonian explicitly depends on time. The approximation methods considered are splitting methods, where the time variable is split off and advanced separately, and commutatorfree Magnustype methods. The timesteps are chosen adaptively based on asymptotically correct estimators of the local error in both cases. It is found that splitting methods are more efficient when the Hamiltonian naturally suggests a separation into kinetic and potential part, whereas Magnustype integrators excel when the structure of the problem only allows to advance the time variable separately.
Introduction
We study systems of linear ordinary differential equations of Schrödinger type
with a timedependent Hermitian matrix \(H :{\mathbb {R}} \rightarrow {\mathbb {C}}^{d \times d}\). The exact flow of (1) is denoted by \({\mathcal {E}}(t,u_0)\) in the following. Highdimensional systems of this form arise for instance in the design of oxide solar cells [31], describing the movement and interaction of electrons within Hubbardtype models of solid state physics, where the explicit timedependence here originates from an external electric field associated with the impact of a photon, or as typical semiclassical models arising in quantum control [40]. In the former application, the computational challenge results from the high dimension of the resulting system. Indeed, for a model with n discrete locations, the state space has dimension \(4^n\). Thus, for a model with the claim of physical relevance, the problem quickly reaches the limitations of modern supercomputers. In the latter application, the semiclassical parameter is chosen as very small, which mandates a fine spacial discretization, again resulting in very large systems of ODEs. Thus the main motivation for the present study is to identify the computationally most efficient numerical time integrators for the considered problem class in order to make largescale simulations of high accuracy feasible on current computer hardware.
The aim of this paper is to compare two approaches to the numerical time integration of problems of the form (1). Popular integrators for timedependent linear homogeneous differential equations are based on the Magnus expansion [23, 35], or on commutatorfree exponentialbased integrators [1]. These have been found to excel over classical Magnus integrators (introduced as numerical methods in [21]) for example in [3] and will therefore be used in the present study. In contrast, nonautonomous problems can also be solved by interpreting the independent variable t as a separate component, which in splitting methods can be frozen over a timestep and propagated separately. This approach is discussed extensively in [16,17,18, 20, 39] and references therein. The success of the splitting approach critically depends on the structure of the underlying problem. If the operator H(t) naturally suggests a splitting, where the timedependent part is cheap to compute for fixed t, this may offer computational advantages when t is propagated along with one suboperator. However, if only t is split off, the required number of compositions in a splitting approach may be prohibitive from a computational point of view. Also, if H has a special structure which can be exploited to increase the efficiency, the introduction of the additional variable t may destroy this structure [16]. We will corroborate these general observations on a number of practical examples, see also [19] for an abstract discussion of the computational effort.
The present comparisons involve methods that have been used in previous studies, but not in comprehensive assessment of the efficiency when applied to a number of applicationmotivated examples. The efficiency of adaptive splitting methods has been studied by the authors for instance in [2, 9], and adaptive Magnustype methods are discussed in [3, 7]. This work adds the aspect of understanding adaptive splitting and Magnustype methods as to their applicability and respective merits. By providing a meticulous comparison on several significant examples from applications, we give a balanced account of advantages and disadvantages of the two numerical approaches. The Hubbard model is of high interest in solid state physics, and therefore a search for the best numerical approach among several contenders is of relevance.
In Sect. 2 of this manuscript, we specify the model model problems that we will subsequently resort to in our comparisons, in Sect. 3 we briefly recapitulate the numerical approaches that are used, and in Sect. 4 we give the results of our numerical comparisons. The main criterion to assess the computational efficiency is the required CPU time to reach a prescribed accuracy, as the considered numerical approaches are fundamentally different in their structure and do not readily admit other metrics.
Model Problems
We consider a Rosen–Zener model related to quantum optics, a Hubbard model of the impact of light on a solid, and a semiclassical problem typical for quantum control.
Rosen–Zener problem As the first example, we consider a Rosen–Zener model from [19], which appears in quantum optics, see also [33]. The associated Schrödinger equation in the interaction picture is given by (1) with
where the initial condition is chosen as \(\psi (0)=(1,\dots ,1)^T.\) The integration interval is \(t \in [5,5]\).
Hubbard model for solar cells Next, we consider a Hubbard model describing the movement and interaction of electrons within an oxide solar cell [25, 31] built from \({{\mathrm {LaVO}}}_3\), with^{Footnote 1}\(H(t)\in {\mathbb {C}}^{4900\times 4900}\). The explicit timedependence here originates from an external electric field associated with the impact of a photon.
This model is given by a finitedimensional Hamiltonian in second quantization of the form
Here, the annihilation and creation operators \(c_{i \sigma }^{}\) and \(c_{j \sigma }^\dagger \) take an electron away from site i with spin \(\sigma \in \{\uparrow ,\downarrow \}\) and add it on site j.
The impact of a photon exciting the system out of equilibrium can be described by a classical electric field pulse, which introduces timedependence to the Hamiltonian (3), see [25]. We choose \({\mathrm {e}}^{{\mathrm {i}} \,\omega (t)}\) with \(\omega (t) =\tfrac{1}{10} \exp \left( \tfrac{1}{6} (t6)^2\cos \left( \tfrac{7\pi }{4}(t6)\right) \right) \), which appears in offdiagonal entries of H(t) depending on the geometry underlying the model of the investigated solid. The model is described in detail in [27].
The oscillating and quickly attenuating electric field generated by the external potential in this model makes adaptive timestepping a relevant issue. Time integration proceeds on the interval \(t\in [0,30]\).
Quantum control A model typical for quantum control of atomic systems which is discussed in [28], see also [40], introduces a potential which explicitly depends on time,
with V(x, t) and the initial condition chosen as
where \(x_0=0.3\), \(k_0=0.1\), \(\delta =10^{3}\) and \(\varepsilon \) in (4a) assumes the values \(2^{6},\ 2^{8},\ 2^{10},\ 2^{12}\). The spatial interval \([1,1]\) is discretized using a Fourier pseudospectral method at 2048 points for periodic boundary conditions. The computation terminates at \(t_{{\mathrm {end}}}=0.75.\)
Adaptive Time Integration
Splitting Methods
Splitting methods constitute a popular divideandconquer approach for numerical time integration of (1) when the Hamiltonian is partitioned, i.e., \({\mathrm {i}}H(t) = A(t) + B(t)\) and the operators A and B have different properties which promise computational advantages when propagated independently. This is for instance typical for the splitting of a Schrödinger operator into kinetic and potential part.
In our context, we will use splitting methods by making the problem formally autonomous by considering t as an additional solution component and adding the equation \(t'=1\). In this setting, time can be advanced separately, or simultaneously with one suboperator if this is autonomous. More precisely, in the definition of the splitting, the operators become
The same holds mutatis mutandis when t is propagated together with B.
For autonomous problems, splitting methods have the following form: At the (time)semidiscrete level, sstage exponential splitting methods use multiplicative combinations of the partial flows \( {\mathcal {E}}_A(t,u_0):u_0 \mapsto u(t)\) with \(u'(t)=A(u(t)),\ u(t_0)=u_0\), and \( {\mathcal {E}}_B(t,u_0):u_0 \mapsto u(t) \) with \(u'(t)=Bu(t),\ u(t_0)=u_0\). For a single step \( (t_0,u_0) \mapsto (t_0+h,u_1) \) with timestep \( t=h \), this reads
where the coefficients \( a_j,b_j,\, j=1 \ldots s \) are determined from order conditions to achieve a desired order of consistency [23].
Local Error Estimators for Splitting Methods
As the basis for adaptation of the timesteps, three classes of local error estimators are used in this study. These have different advantages depending on the context in which they are applied [5].

(i)
Embedded pairs of splitting formulae have first been considered in [32] and are based on reusing a number of evaluations from the basic integrator. In this paper, we will focus on the pairs [6, Emb 4/3 AK p] of orders four and three and [6, Emb 5/4 AK (ii)] of orders five and four, which were found to be the most successful in earlier work [2].

(ii)
A defectbased error estimator has been proposed and analyzed in [8, 10,11,12]. In order to construct an error estimator associated with a splitting method of order \(p \ge 1\), an integral representation of the local error involving the defect \({\mathcal {D}}\) of the numerical approximation is evaluated by means of an Hermite quadrature formula. Due to the fact that the validity of the pth order conditions ensures that the first \(p1\) derivatives of \({\mathcal {D}}\) vanish at \(t=t_0\), this leads to a local error estimator involving a single evaluation of the defect,
$$\begin{aligned} {\mathcal {P}}(t,u_0) = \tfrac{1}{p+1}\,t\,{\mathcal {D}}(t,u_0) \,\approx \, {\mathcal {L}}(t,u_0) = {\mathcal {S}}(t,u_0){\mathcal {E}}(t,u_0)\,. \end{aligned}$$(6)This device works generally for splittings of any order into an arbitrary number of operators if Fréchet derivatives of the subflows are available, see [4]. We use the defectbased error estimator in conjunction with the integrators in [6, Emb 4/3 AK p] and [6, Emb 5/4 AK (ii)], since these are close to optimal.

(iii)
For adjoint pairs of formulae of odd order p, an asymptotically correct error estimator can be computed at the same cost as for the basic method, see [5]. Since the error estimator is easy to construct and evaluate in this case, we employ the pair [6, PP 5/6 A] of orders 5/6, since this was found to be efficient for high accuracy demands for instance in [2]. We will also employ this optimized method in conjunction with the defectbased error estimator for reasons of comparison.
All the error estimates we use in our comparisons are asymptotically correct, i.e., the deviation of the error estimator from the true error tends to zero faster than does the error.
Commutatorfree Magnustype Integrators
A successful and much used class of integration methods is given by higherorder commutatorfree Magnustype integrators (CFM) [1, 22]. These approximate the exact flow in terms of products of exponentials of linear combinations of the system matrix evaluated at different times, avoiding evaluation and storage of commutators. These have been found to excel over classical Magnus integrators in applications in our interest in [3].
One step of a CFM scheme for (1) starting at \((t_0,u_0)\) is defined by^{Footnote 2}
where the coefficients \(a_{jk}\), \(c_k\) are determined from the order conditions (a system of polynomial equations in the coefficients) such that the method attains convergence order p, see for example [19] and references therein. Algorithms to efficiently generate the order conditions are described for instance in [26]. Since such a system of equations generally does not define a unique solution, numerical optimization techniques are employed, for example minimizing the leading local error term of the resulting integrator.
In this study, we will use the methods referred to as CF4oH and CF6n in [3].
The choice of the methods above for our comparisons is motivated by the fact that these two methods were found to be the most efficient CFM methods in the study [3].
Local Error Estimation for Magnustype Methods
As a basis for adaptive timestepping, defectbased error estimators for CFM methods and for classical Magnus integrators have been introduced in [7]. For the defect
it holds that
for an order p method.
The local error \({\mathcal {L}}(\tau )\psi _0:=({\mathcal {S}}(\tau ;t_0){\mathcal {E}}(\tau ;t_0))\psi _0\) can be expressed via the variationofconstant formula as
For the practical evaluation of the defect, the derivative of matrix exponentials of the form
is required. The function \({\varGamma }\) can be expressed as an infinite series or alternatively as an integral. These are approximated by truncation or numerical Hermite quadrature, respectively, to yield a computable quantity \({{\tilde{{\varGamma }}}}\) and an approximate defect \({\tilde{{\mathcal {D}}}}\). The resulting computable error estimator is denoted by \({\tilde{{\mathcal {P}}}}\). The asymptotical correctness of the error estimators was established in [7].
In the numerical experiments reported in Sect. 4 below, truncation of the Taylor expansion has been used throughout.
Adaptive Lanczos Method
The crucial computational step in any of the Magnustype methods described above, is the evaluation of the action of a matrix exponential
Note that this subproblem is also solved in the splitting approximation of the problems (2) and (3), whereas in (4), the pseudospectral space discretization renders this substep the trivial exponentiation of a diagonal matrix. The standard Krylov approximation to \({\mathrm {e}}^{{\mathrm {i}}t {\varOmega }} v\) reads
with \(T_m=(\tau _{i,j})\) tridiagonal and \(V_m\) an orthonormal basis of the Krylov space \({\mathcal {K}}_m({\varOmega },v) = {\mathrm {span}}\{v,{\varOmega }v,\ldots ,{\varOmega }^{m1}v\} \subseteq {\mathbf {C}}^n\). For Hermitian or skewHermitian matrices \({\varOmega }\), the Lanczos method [36] constitutes a computationally efficient realization.
In [30], a timestepping strategy was introduced which is based on the defect of the approximation. Due to the success of this strategy documented ibidem, we use it invariantly in the Magnustype integrators. The asymptotically correct error estimator is based on the defect operator
The local error operator \(L_m(t)=E(t)S_m(t)\) can be represented as
Numerical quadrature applied to this defectbased integral representation yields a computable, asymptotically correct local error bound satisfying (see [30]),
with \(\gamma _m= \prod _{j=1}^{m1} (T_m)_{j+1,j}.\)
As an error tolerance for the Lanczos matrix exponentiation, we prescribe \(10^{12}\). This allows to realize highly accurate timestepping on the basis of this approximation with tolerance requirements as strict as \(10^{12}\).
Stepsize Selection
Based on a local error estimator, the time stepsize is adapted such that the tolerance is expected to be satisfied in the following step. If \(h_{\text {old}}\) denotes the present stepsize, the next stepsize \(h_{\text {new}}\) in an order p method is predicted as (see [24, 37])
where we choose the parameters as \( \alpha = 0.9 \), \( \alpha _{\text {min}} = 0.25 \), \( \alpha _{\text {max}} = 4.0 \), and \({\mathcal {P}}(h_{\text {old}})\) is an asymptotically correct estimator for the local error arising in the previous timestep. This established and widely used strategy incorporates safety factors to avoid an oscillating and unstable behavior.
Numerical Results
Here, we give the results of our experimental comparisons of the numerical methods described in Sect. 3. The numerical results have been obtained based on implementations which can be found at
https://github.com/HaraldHofstaetter/TimeDependentLinearODESystems.jl and https://github.com/HaraldHofstaetter/TSSM.jl.
As a measure of computational efficiency, we resort to CPU time on the Vienna Scientific Cluster. Its third generation cluster VSC3 has 2020 nodes, each equipped with 2 processors (Intel Xeon E52650v2, 2.6 GHz, 8 cores). The runtimes we give below are averages over 100 identical runs on a single compute node, respectively. Runtime seems to be the most reasonable measure of computational efficiency due to the very different nature of the two numerical approaches. Two different local error tolerances \(10^{5}\) and \(10^{9}\) are prescribed for all examples, for (4) the tolerance \(10^{12}\) could also be reached.
Rosen–Zener model. In Table 1 we show the results for the Rosen–Zener model (2). For the splitting methods, only the time variable is split off and the Hamiltonian is exponentiated as a whole. In modern computer arithmetics, a conceivable splitting into real and imaginary part does not promise a computational advantage. We observe that the most efficient exponentialbased method is CF4oH, while Emb 4/3 AK p is the best splitting method for the larger tolerance \(10^{5}\), and PP 5/6 A excels for tolerance \(10^{9}\). Note that the number of timesteps does not immediately correspond with the computational effort, the commutatorfree Magnustype method of order six requires the fewest steps, but is more expensive in each step and thus not the fastest integrator. The fastest exponentialbased integrator is almost twice as fast as the best splitting method.
Hubbard model. For the Hubbard model of solar cells (3) we obtain a similar picture. Again, only the time variable is split off. Table 2 shows the runtimes for tolerances \(10^{5}\) and \(10^{9}\). The fourth order commutatorfree Magnustype integrator CF4oH is the most efficient for both tolerances, and again, Emb 4/3 AK p is the best splitting method for the larger tolerance, and PP 5/6 A for the stricter tolerance. The best exponentialbased method again excels over the best splitting method.
Quantum control. The results for the semiclassical problem (4) show a different picture than the previous investigations. The reason is obvious: The problem (4) suggests a natural splitting into kinetic and potential part, and hence t can be propagated efficiently alongside with the autonomous kinetic operator. We vary \(\varepsilon \) from \(\varepsilon =2^{6}\) to \(\varepsilon =2^{12}\) in Tables 3, 4, 5 and 6. For this example, a tolerance of \(10^{12}\) could additionally be achieved and is added to the numerical results. Throughout, the best splitting method is EMB 5/4 AK (ii), and the best Magnustype method is CF4oH. For larger \(\varepsilon \), splitting methods are clearly to be preferred, but this advantage is significantly diminished for the more oscillatory problems for smaller \(\varepsilon \). Indeed, Magnustype integrators are known to excel for oscillatory problems. For larger \(\varepsilon \) and particularly larger tolerances, Emb 5/4 AK (ii) is by far more efficient than the best exponentialbased method, but for smaller \(\varepsilon \), this advantage is dimished, and for \(\varepsilon =2^{10}\) and \(2^{12}\) and tolerance \(10^{12}\), CF4oH is even slightly faster. The reason may be the additional splitting error which contributes to dimished efficiency due to reduced accuracy for a given computational effort.
Conclusions
We have studied the differences between two fundamentally diverse approaches for the solution of linear nonautonomous systems of differential equations. Exponentialbased methods related to the Magnus expansion are contrasted with splitting methods, where the time variable is split off and suitably propagated. Both approaches allow to construct asymptotically correct estimators for the local timestepping error and implement adaptive timestepping on this basis. Which method is more efficient depends on the problem structure. If only the time variable is split off, the additional substeps induced in the splitting procedure seem not to be justified from the point of view of computational efficiency. However, if the problem naturally suggests a splitting into a timedependent and a timeindependent part, the approach may be more efficient. However, for highly oscillatory problems, the splitting error is too large and Magnustype integrators are again to be preferred. Our findings may also have an impact on the study of timedependent differential equations of other classes such as differentialalgebraic equations [15], functional and stochastic differential equations [34], or fractional differential equations [29, 38], see also [13, 14].
Notes
The dimension of the matrix in this model grows exponentially with the number of considered sites in the Hubbard model of the solid, making the issue of an efficient time integrator crucial. For our illustrations in this paper, we choose a model of manageable size.
Note the slight difference in notation as compared to splitting methods, which is motivated by the fact that for timedependent linear problems, the evolution depends on the initial time and represents a linear operator applied to the initial value.
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Auzinger, W., Hofstätter, H., Koch, O. et al. Adaptive Time Propagation for Timedependent Schrödinger equations. Int. J. Appl. Comput. Math 7, 6 (2021). https://doi.org/10.1007/s40819020009379
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DOI: https://doi.org/10.1007/s40819020009379
Keywords
 Timedependent Schrödinger equations
 Splitting methods
 Magnustype integrators
 Adaptive stepsize selection