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, in this paper 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:清晰的分级分级加压方法最近被证明成功地整合了高维部分差异方程式(PDEs)的初步价值问题。然而,非线性和僵硬的结合可能会引入时间步骤限制,使得明显集成在计算上是行不通的。为了克服这一问题,我们在本文件中开发了一种新的隐含的分级适应算法类别,用于将非线性进化方程式在高压元体上的时间整合。这些算法的基础是与常规的分步制方案执行一个时间步骤,随后是隐含的固定分点转动步骤,涉及在高压方程式上进行分级调解动操作。不直线的分级分级脱轨方法可以直接实施,因为它们只依赖在数个数方程式之间的算操作,而这种算法可以通过高效和可伸缩的平行的平行算法进行。为Allen-Cahn方程式、Fokker-Planc 方程式和非线性Schr\\\oder等方程式提供和讨论显示隐性分解效果的数值应用。