Lie–Trotter Matrix Splitting

Worked Example

Consider the matrices:

A = [[0, 1], [0, 0]], B = [[0, 0], [-1, 0]]

Here s = -1. Since A² = B² = 0, their exact exponentials are simplyexp(hA) = I + hA and exp(hB) = I + hB. We want to approximate the combined flow at t = 1 using n = 1 step of the Lie-Trotter splitting:

1. Set step size h = 1 / 1 = 1.
2. Evaluate exp(hA) = [[1, 1], [0, 1]].
3. Evaluate exp(hB) = [[1, 0], [-1, 1]].
4. Multiply: exp(hA)exp(hB) = [[1, 1], [0, 1]] * [[1, 0], [-1, 1]] = [[0, 1], [-1, 1]].

The true coupled flow for A + B = [[0, 1], [-1, 0]] is a rotation matrix[[cos(1), sin(1)], [-sin(1), cos(1)]] ≈ [[0.5403, 0.8415], [-0.8415, 0.5403]]. The difference between the approximation and the exact matrix is the splitting error.

Experiment

You can interact with these subflows using the module:

1. Open the Lie–Trotter Matrix Splitting module.
2. Set Splitting to "Lie Trotter" and Steps to 4. Observe the finite-step error.
3. Change Splitting to "Strang" (the symmetric exp(hA/2)exp(hB)exp(hA/2)) and notice how the Frobenius error decreases significantly, demonstrating its higher global order.
4. Change Lower generator sign to 1 (making it a hyperbolic flow rather than a rotation) and observe the new trajectories.
5. Change Lower generator sign to 0 (where A and B commute) and see that the error drops to 0 immediately.

Limitations

This visualization only considers the specified exact cases (rotations, hyperbolic flows, and shear) on a finite 2D domain. Real-world operator splitting is typically applied to large sparse matrices (like discretized PDEs) where computing exp(t(A+B)) is intractably expensive, but solving A and Bseparately is highly efficient. Furthermore, higher-order commutators and Baker-Campbell-Hausdorff expansions govern the exact error bounds for larger time steps.

References

For more information on splitting methods and their applications to differential equations:

• Computational Physics - Operator Splitting