Numerical Solution to Coupled Burgers’ Equations by Gaussian-Based Hermite Collocation Scheme

Nissaya Chuathong,Sayan Kaennakham

doi:10.1155/2018/3416860

Abstract

One of the most challenging PDE forms in fluid dynamics namely Burgers equations is solved numerically in this work. Its transient, nonlinear, and coupling structure are carefully treated. The Hermite type of collocation mesh-free method is applied to the spatial terms and the 4th-order Runge Kutta is adopted to discretize the governing equations in time. The method is applied in conjunction with the Gaussian radial basis function. The effect of viscous force at high Reynolds number up to 1,300 is investigated using the method. For the purpose of validation, a conventional global collocation scheme (also known as “Kansa” method) is applied parallelly. Solutions obtained are validated against the exact solution and also with some other numerical works available in literature when possible.

Highlights

Fluid is known to dominate most part of the planet and can be modelled mathematically by the well-known NavierStoke equation
With Hermite interpolation technique, it begins with writing the approximation of solution for ũ(x) andV(x), respectively, with the same radial basis function φ(‖x − xj‖2), as follows: u (x, t) ≃ ũ (x, t)
A further developed version of Kansa collocation method called “Hermite” scheme was applied to Average RMS error

Summary

Introduction

Fluid is known to dominate most part of the planet and can be modelled mathematically by the well-known NavierStoke equation. Acknowledged as a simple case of the Navier-Stoke equation, the famous “Burgers” equations, a system of equations that describes the interaction between two crucial physical manners of nature, convection, and diffusion, are used to model variety of applications. Amongst the well-known numerical scheme, finite volume, finite difference, and finite element method that have been invented, developed, and applied in a wide range of science and engineering problems, a rather young idea was discovered and has been categorized as “meshless/mesh-free” methods. The methods under this category have recently become promising alternative tools for numerically solving variety of science and engineering problems.

The Gaussian-Based Collocation Meshless Method

Implementation to Burgers’ Equations

Numerical Experiments and General Discussion

Conclusion