Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

A.S.V Ravi Kanth,Deepika Deepika

doi:10.1051/matecconf/20165705004

Abstract

In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.

Highlights

We present an implicit non-polynomial spline functions based scheme for the numerical solution of the following one dimensional convection-diffusion equation, ut Hux J uxx g(x,t), 0 d x d 1, t t 0 (1)
It arises in various fields of applied sciences and engineering such as oil reservoir simulations, transport of mass and energy, global weather production, dispersion of diffusion process, due to these vide variety of physical implementations a great deal of researches have been done for the numerical and closed form solutions of convection diffusion type equation for eg., Jain and Aziz [4] used the adaptive spline function approximation for the numerical solution of convection-diffusion equation
The stability has been proven through the Von Neumann technique,the numerical solution can be expressed by means of a Fourier series

Summary

Introduction

We present an implicit non-polynomial spline functions based scheme for the numerical solution of the following one dimensional convection-diffusion equation, ut Hux J uxx g(x,t), 0 d x d 1, t t 0 (1). The convection-diffusion equation describes the various physical phenomena where energy is transformed inside a physical system due to the combination of convection as well as diffusion processes It arises in various fields of applied sciences and engineering such as oil reservoir simulations, transport of mass and energy, global weather production, dispersion of diffusion process, due to these vide variety of physical implementations a great deal of researches have been done for the numerical and closed form solutions of convection diffusion type equation for eg., Jain and Aziz [4] used the adaptive spline function approximation for the numerical solution of convection-diffusion equation. Mohammadi [5] applied exponential B-spline collocation method for solving convection-diffusion equation with Dirichlet’s type boundary conditions. Mohammadi [2] presented the exponential spline approach and Lin [3]

Derivation of the Numerical Scheme

Stability analysis

Numerical results

Conclusion