Explicit step-truncation tensor methods have recently proven successful in integrating initial value problems for high-dimensional partial differential equations (PDEs). However, the combination of non-linearity and stiffness may introduce time-step restrictions which could make explicit integration computationally infeasible. To overcome this problem, we develop a new class of implicit rank-adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms are based on performing one time step with a conventional time-stepping scheme, followed by an implicit fixed point iteration step involving a rank-adaptive truncation operation onto a tensor manifold. Implicit step truncation methods are straightforward to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Numerical applications demonstrating the effectiveness of implicit step-truncation tensor integrators are presented and discussed for the Allen-Cahn equation, the Fokker-Planck equation, and the nonlinear Schr\"odinger equation.

翻译：显式步长截断张量方法近期在求解高维偏微分方程（PDEs）的初值问题上取得了成功。然而，非线性与刚性的结合可能引入时间步长限制，使得显式积分的计算变得不可行。为解决这一问题，我们提出了一类新的隐式秩自适应算法，用于在张量流形上对非线性演化方程进行时间积分。这些算法基于以下步骤：首先采用传统时间推进格式完成一个时间步的计算，随后执行一个涉及秩自适应截断操作（将结果投影到张量流形上）的隐式不动点迭代步骤。隐式步长截断方法易于实现，因为它们仅依赖于张量间的算术运算，而此类运算可通过高效且可扩展的并行算法完成。我们通过数值应用（包括Allen-Cahn方程、Fokker-Planck方程和非线性薛定谔方程）展示并讨论了隐式步长截断张量积分器的有效性。