On Runge-Kutta methods of order 10
A family of explicit 15-stage Runge-Kutta methods of order 10 is derived.
math.NA↗
arXiv subjects
Misha Stepanov
Publications and source records attributed to Misha Stepanov.
Lagrangian Attention Tensor Networks for Velocity Gradient Statistical Modeling
Direct numerical simulation of turbulence at realistic Reynolds numbers is still beyond current computational capability, necessitating models that reduce the number of resolved spatial scales. Motivated by phenomenology and recent data-driven works based on universality of the smallest scales in fully developed turbulence, the statistical dynamics of the velocity gradient tensor (VGT) at the Kolmogorov scale become of critical importance in advancing turbulence models. Physics-informed machine learning has found considerable success in exploiting large datasets taken from direct numerical simulation of Navier-Stokes to improve models for the evolution of the VGT. In this work, we follow the long line of blending physical insight with data analysis to simultaneously advance both the modeling and understanding of the phenomenology of the VGT. Using the intimate connection between VGT evolution and fluid deformation, we develop the Lagrangian attention tensor network approach that significantly improves over current physics-informed machine learning methods. We demonstrate state-of-the-art performance in both a-priori and a-posteriori metrics, before interpreting the trained attention mechanisms to discover a surprising connection between the history of the strain-rate-tensor and the pressure Hessian.
physics.flu-dyn↗
Eight-stage pseudo-symplectic Runge-Kutta methods of order (4, 8)
Using simplifying assumptions that are related to the time reversal symmetry, a 1-dimensional family of 8-stage pseudo-symplectic Runge-Kutta methods of order (4, 8), i.e., methods of order 4 that preserve symplectic structure up to order 8, is derived. An example of 7-stage method of order (4, 9) is given.
math.NA↗
Two continuous (4, 5) pairs of explicit 9-stage Runge-Kutta methods
An 11-dimensional family of embedded (4, 5) pairs of explicit 9-stage Runge-Kutta methods with an interpolant of order 5 is derived. Two optimized for efficiency pairs are presented.
math.NA↗
Embedded (4, 5) pairs of explicit 7-stage Runge-Kutta methods with FSAL property
The general case of embedded (4, 5) pairs of explicit 7-stage Runge--Kutta methods with FSAL property (a_7j = b_j, 1 <= j <= 7, c_7 = 1) is considered. Besides exceptional cases, the pairs form five 4-dimensional families. The pairs within two (already known) families satisfy the simplifying assumption sum_j a_ij c_j = c_i^2 / 2, i >= 3.
math.NA↗
Most probable failure scenario in a model power grid with random power demand
We consider a simple system with a local synchronous generator and a load whose power consumption is a random process. The most probable scenario of system failure (synchronization loss) is considered, and it is argued that its knowledge is virtually enough to estimate the probability of failure per unit time. We discuss two numerical methods to obtain the "optimal" evolution leading to failure.
math.OC↗
Instantons causing iterative decoding to cycle
It is speculated that the most probable channel noise realizations (instantons) that cause the iterative decoding of low-density parity-check codes to fail make the decoding not to converge. The Wiberg's formula is generalized for the case when the part of a computational tree that contributes to the output at its center is ambiguous. Two methods of finding the instantons for large number of iterations are presented and tested on Tanner's [155, 64, 20] code and Gaussian channel. The inherently dynamic instanton with effective distance of 11.475333 is found.
cs.IT↗