More precisely, Pasch [118
] found that that an exponential integrator worked well when the flow’s Jacobian matrix was
computed exactly (using the symbolic-differentiation technique described in Section 8.5.4). However, when the Jacobian
matrix was approximated using the numerical-perturbation technique described in Section 8.5.4, Pasch found that the
pseudo-time integration became unstable at high numerical resolutions. Pasch [118
] also notes that the exponential integrator
uses a very large amount of memory.