Runge-Kutta methods#

Runge-Kutta methods use a Taylor expansion of the state to approximate

\[\psi_{k+1} = \psi_k + \sum_{p}\frac{1}{p!}(-\Delta \beta H)^p\frac{\partial^{(p)} \psi}{\partial \beta^{(p)}}.\]

\(\Delta \beta\) can either be real or imaginary time, and hence these methods are suitable for both evolution and optimization problems. When using MPS, these methods perform a global optimization of the MPS at each step, i.e., they update all tensors simultaneously.

The order of the expansion p determines the truncation error of the method, which is \(O(\Delta \beta ^{p+1})\), and also the cost of the method, since a higher order implies more operations. Thus, it is important to consider trade-off in cost and accuracy to choose the most suitable method for each application.

The SeeMPS library considers four methods.

1. Euler method#

This is an explicit, first-order Taylor approximation of the evolution, with an error \(\mathcal{O}(\Delta\beta^2)\) and a simple update with a fixed time-step \(\beta_k = k \Delta\beta\).

\[\begin{split}\psi_0 &= \psi(\beta_0), \\ \psi_{k+1} &= \psi_k - \Delta\beta H \psi_k, \quad \text{for } k=0,1,\dots,N-1.\end{split}\]

2. Improved Euler or Heun method#

This is a second-order, fixed-step explicit method that uses two matrix-vector multiplications and two linear combinations of vectors to achieve an error \(\mathcal{O}(\Delta\beta^3)\).

\[\begin{split}\psi_{k+1} = \psi_k - \frac{\Delta\beta}{2} \left[v_1 + H(\psi_k - \Delta\beta v_1)\right], \\ \text{with } v_1 = H \psi_k.\end{split}\]

3. Fourth-order Runge-Kutta method#

This algorithm achieves an error \(\mathcal{O}(\Delta\beta^5)\) using four matrix-vector multiplications and four linear combinations of vectors.

\[\begin{split}\psi_{k+1} &= \psi_k + \frac{\Delta\beta}{6}(v_1 + 2v_2 + 2v_3 + v_4), \\ v_1 &= -H \psi_k, \\ v_2 &= -H\left(\psi_k + \frac{\Delta\beta}{2}v_1\right), \\ v_3 &= -H\left(\psi_k + \frac{\Delta\beta}{2}v_2\right), \\ v_4 &= -H\left(\psi_k + \Delta\beta v_3\right).\end{split}\]

4. Runge-Kutta-Fehlberg method#

The Runge-Kutta-Fehlberg algorithm is an adaptative step-size solver that combines a fifth-order accurate integrator \(O(\Delta\beta^5)\) with a sixth-order error estimator \(O(\Delta\beta^6)\). This combination dynamically adjusts the step size \(\Delta\beta\) to maintain the integration error within a specified tolerance. The method requires an initial estimate of the step size, which can be obtained from a simpler method. Each iteration involves six matrix-vector multiplications and six linear combinations, and it may repeat the evolution steps if the proposed step size is deemed unsuitable.

`euler`(H, t_span, state[, steps, strategy, ...])	Solve a Schrodinger equation using the Euler method.
`euler2`(H, t_span, state[, steps, strategy, ...])	Solve a Schrodinger equation using the 2nd order Euler method.
`runge_kutta`(H, t_span, state[, steps, ...])	Solve a Schrodinger equation using a fourth order Runge-Kutta method.
`runge_kutta_fehlberg`(H, t_span, state[, ...])	Solve a Schrodinger equation using a fourth order Runge-Kutta method.