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

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.

Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

“…The following results can be concluded from Table 7 : MC D-matrices are more efficient and accurate than the method used in Akram and Beck ( 2015 ) as a direct method without transformation. By transforming the given problem into seven 1st order ODEs, MC D-matrices are still more efficient and accurate than the bvp5c Matlab function.…”

“…Consider the 7th ODE (Akram and Beck 2015 ): subject to: with exact solution: , where, y is particles’ velocity for a limited time.…”

A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

“…Example Consider the following seventh‐order IVP 1,22 : with the exact solution: v ( t ) = e t (1 − t ) t . Note that v ( t ) represents the differences of the velocity at different scales in the problem of turbulent flows.…”

“…IVPs arise in many fields and applications and specially in physics and engineering. For example, the hierarchical model which appears in physics can be modeled by a seventh‐order IVP 1 . Moreover, the governor equations in the fluid dynamics are nonlinear IVPs or IBVPs 2 .…”

Approximating Real-Life BVPs via Chebyshev Polynomials’ First Derivative Pseudo-Galerkin Method

“…BVPs are used to model various problems in some fields, such as economics, biology, and engineering [1][2][3][4][5]. Due to the importance of ODEs, significant research work has been carried out about these problems [6][7][8][9][10][11]. In most instances, the exact solution of some ODEs cannot be obtained analytically, and numerical methods are considered as the way to obtain it.…”