  We develop new adaptive algorithms for temporal integration of nonlinear
evolution equations on tensor manifolds. These algorithms, which we call
step-truncation methods, are based on performing one time step with a
conventional time-stepping scheme, followed by a truncation operation onto a
tensor manifold. By selecting the rank of the tensor manifold adaptively to
satisfy stability and accuracy requirements, we prove convergence of a wide
range of step-truncation methods, including explicit one-step and multi-step
methods. These methods are very easy to implement as they rely only on
arithmetic operations between tensors, which can be performed by efficient and
scalable parallel algorithms. Adaptive step-truncation methods can be used to
compute numerical solutions of high-dimensional PDEs, which have become central
to many new areas of application such optimal mass transport, random dynamical
systems, and mean field optimal control. Numerical applications are presented
and discussed for a Fokker-Planck equation with spatially dependent drift on a
flat torus of dimension two and four.
