Implicit Integration of Nonlinear Evolution Equations on Tensor Manifolds

Abstract Explicit step-truncation tensor methods have recently proven successful in integrating initial value problems for high-dimensional partial differential equations. 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 implicit rank-adaptive algorithms temporal nonlinear evolution equations on manifolds. These are based performing one time step with conventional time-stepping scheme, followed by an fixed point iteration involving truncation operation onto manifold. Implicit straightforward to implement as they rely only arithmetic operations between tensors, can be performed efficient scalable parallel algorithms. Numerical applications demonstrating effectiveness integrators presented discussed Allen–Cahn equation, Fokker–Planck Schrödinger equation.