Tensor Train-based cross interpolation method for solving high-dimensional PDF transport equation of turbulent flows

Solving high-dimensional partial differential equations (PDEs) encountered in modeling probability density functions (PDFs) for turbulent reacting flows suffers from the curse of dimensionality. To address this issue, we introduce a methodology based on our recently developed tensor interpolation algorithm for the time integration of PDF transport equations on the low-rank tensor train manifolds. The approach achieves near-optimal computational savings both in terms of memory and floating-point operations by applying a cross algorithm based on the discrete empirical interpolation method (DEIM), which selects only a small subset of the tensor entries to compute low-rank updates. The method is robust and stable even when the system has small singular values, which usually cause numerical instability. The time integration approach is extended to high-order explicit Runge–Kutta schemes. The algorithm is straightforward to implement. It only needs evaluations of the full-order model at strategically chosen entries and does not rely on tangent space projections, which are often intrusive for efficient implementation.