Hierarchical cascade model leading to 7-th order initial value problem

TL;DR

This paper introduces a hierarchical cascade model for turbulent flows that results in a high-order initial value problem, and develops a non-polynomial spline method to solve it efficiently with high precision.

Contribution

The paper formulates a 7th order initial value problem from a hierarchical cascade model and proposes an optimized spline-based numerical solution method.

Findings

The method achieves a truncation error of order O(h^5).

Numerical examples demonstrate the effectiveness of the approach.

The model captures hierarchical dynamics in turbulence.

Abstract

In turbulent flows, local velocity differences often obey a cascade-like hierarchical dynamics, in the sense that local velocity differences at a given scale k are driven by deterministic and random forces from the next-higher scale k-1. Here we consider such a hierarchically coupled model with periodic boundary conditions, and show that it leads to an N-th order initial value problem, where N is the number of cascade steps. We deal in detail with the case N=7 and introduce a non-polynomial spline method that solves the problem for arbitrary driving forces. Several examples of driving forces are considered, and estimates of the numerical precision of our method are given. We show how to optimize the numerical method to obtain a truncation error of order O(h^5) rather than O(h^2), where h is the discretization step.