  We consider the numerical computation of finite-genus solutions of the
Korteweg-de Vries equation when the genus is large. Our method applies both to
the initial-value problem when spectral data can be computed and to dressing
scenarios when spectral data is specified arbitrarily. In order to compute
large genus solutions, we employ a weighted Chebyshev basis to solve an
associated singular integral equation. We also extend previous work to compute
period matrices and the Abel map when the genus is large, maintaining numerical
stability. We demonstrate our method on four different classes of solutions.
Specifically, we demonstrate dispersive quantization for "box" initial data and
demonstrate how a large genus limit can be taken to produce a new class of
potentials.
