  Digital quantum simulation relies on Trotterization to discretize time
evolution into elementary quantum gates. On current quantum processors with
notable gate imperfections, there is a critical tradeoff between improved
accuracy for finer timesteps, and increased error rate on account of the larger
circuit depth. We present an adaptive Trotterization algorithm to cope with
time-dependent Hamiltonians, where we propose a concept of piecewise
"conserved" quantities to estimate errors in the time evolution between two
(nearby) points in time; these allow us to bound the errors accumulated over
the full simulation period. They reduce to standard conservation laws in the
case of time-independent Hamiltonians, for which we first developed an adaptive
Trotterization scheme [PRX Quantum 4, 030319]. We validate the algorithm for a
time-dependent quantum spin chain, demonstrating that it can outperform the
conventional Trotter algorithm with a fixed step size at a controlled error.
