Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation

Kedir Aliyi Koroche Kedir Aliyi Koroche

doi:10.11648/j.ijamtp.20210702.11

Abstract

In this paper, the piecewise parabolic method is presented for solving the one-dimensional advection-diffusion type equation and its application to the burger equation. First, the given solution domain is discretized by using a uniform Discretization grid point. Next by applying the integration in terms of spatial variable, we discretized the given advection-diffusion type equation and converting it into the system of first-order ordinary differential equation in terms of temporarily variable. Next, by using Taylor series expansion we discretized the obtained system of ordinary differential equation and obtain the central finite difference equation. Then using this difference equation, the given advection-diffusion type equation is solved by using the parabolic method at each specified grid point. To validate the applicability of the proposed method, four model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm L2 and maximum absolute error norm L∞ and the local behavior of the solution is captured exactly. Numerical, exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root means square error norm L2 and maximum absolute error norm L∞ presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving advection-diffusion type equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve the advection-diffusion type equation.

Highlights

The nonlinear advection-diffusion type equation is one of the popular and important models describing many phenomena derived from various areas of mathematical physics and engineering fields [1]
The nonlinear homogenous quasi-linear parabolic partial differential is encountered in the theory of shock waves, mathematical modeling of turbulent fluid, and in continuous stochastic processes [2, 6]
This paper aims to develop a parabolic method that is capable of solving onedimensional advection-diffusion and approximate the exact solution

Summary

Introduction

The nonlinear advection-diffusion type equation is one of the popular and important models describing many phenomena derived from various areas of mathematical physics and engineering fields [1]. The nonlinear model arises in gas dynamics, water waves, electrodynamics, chemical reactions, transport of pollutants flood, and ecological systems [10] This equation is found in the form of hydrodynamics, shock waves, heat conduction [1], and it is called quasi-linear parabolic partial differential [2, 4, 6]. The nonlinear homogenous quasi-linear parabolic partial differential is encountered in the theory of shock waves, mathematical modeling of turbulent fluid, and in continuous stochastic processes [2, 6] Such a type of equation was firstly introduced by Bateman [4] in 1915 and he proposed the steady-state solution of the problem [2]. Burgers’ equation arises in many physical problems including one-dimensional turbulence, sound waves in a viscous medium, shock waves in a viscous medium, waves in fluid-filled viscous elastic tubes, and

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