  Abstract Simulating mixed-state evolution in open quantum systems is crucial
for various chemical physics, quantum optics, and computer science
applications. These simulations typically follow the Lindblad master equation
dynamics. An alternative approach known as quantum state diffusion unraveling
is based on the trajectories of pure states generated by random wave functions,
which evolve according to a nonlinear It\^o-Schr\"odinger equation (ISE). This
study introduces weak first- and second-order solvers for the ISE based on
directly applying the It\^o-Taylor expansion with exact derivatives in the
interaction picture. We tested the method on free and driven Morse oscillators
coupled to a thermal environment and found that both orders allowed practical
estimation with a few dozen iterations. The variance was relatively small
compared to the linear unraveling and did not grow with time. The second-order
solver delivers much higher accuracy and stability with bigger time steps than
the first-order scheme, with a small additional workload. However, the
second-order algorithm has quadratic complexity with the number of Lindblad
operators as opposed to the linear complexity of the first-order algorithm.
