论文标题
在张量流上的非线性演化方程的隐式集成
Implicit integration of nonlinear evolution equations on tensor manifolds
论文作者
论文摘要
明确的步长张量方法最近已证明成功地整合了高维偏微分方程（PDE）的初始值问题。但是，非线性和刚度的组合可能会引入时间步长限制，这可能使明确的集成在计算上不可行。为了克服这个问题，我们开发了一类新的隐式等级自适应算法，用于在张量歧管上的非线性进化方程的时间整合。这些算法基于使用常规的时间稳定方案执行一个时间步骤，然后进行隐式固定点迭代步骤，涉及到张量歧管上的等级自动截断操作。隐式步骤截断方法是直接实现的，因为它们仅依赖于张量之间的算术操作，这可以通过有效且可扩展的并行算法执行。为Allen-Cahn方程，Fokker-Planck方程和非线性Schrödinger方程式提供了表明并讨论隐式台词张量积分器的有效性的数值应用。
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 Schrödinger equation.
