Computational Solutions of Two Dimensional Convection Diffusion Equation Using Crank-Nicolson and Time Efficient ADI

Muhammad Saqib,Daoud Suleiman Mashat,Shahid Hasnain

doi:10.4236/ajcm.2017.73019

Muhammad Saqib, Daoud Suleiman Mashat + Show 1 more

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https://doi.org/10.4236/ajcm.2017.73019

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Abstract

To develop an efficient numerical scheme for two-dimensional convection diffusion equation using Crank-Nicholson and ADI, time-dependent nonlinear system is discussed. These schemes are of second order accurate in apace and time solved at each time level. The procedure was combined with Iterative methods to solve non-linear systems. Efficiency and accuracy are studied in term of L2, L∞ norms confirmed by numerical results by choosing two test examples. Numerical results show that proposed alternating direction implicit scheme was very efficient and reliable for solving two dimensional nonlinear convection diffusion equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

Highlights

In this paper we have extended our previous approach associated to two dimension Convection-diffusion equation
The great Physicist Johannes Martinus Burgers discovered Burgers equation, which is non-linear parabolic partial differential equation (PDE) and widely used as a model in many engineering problems, which explains such as physical flow phenomena in fluid dynamics, turbulence, boundary layer behavior, shock wave formation, and mass transport [1]
We extended our work to enhance our knowledge towards two dimensional Convection diffusion equation.Two test problems were taken to understand the numerical solution with finite difference schemes

Summary

Introduction

In this paper we have extended our previous approach associated to two dimension Convection-diffusion equation. The great Physicist Johannes Martinus Burgers discovered Burgers equation, which is non-linear parabolic partial differential equation (PDE) and widely used as a model in many engineering problems, which explains such as physical flow phenomena in fluid dynamics, turbulence, boundary layer behavior, shock wave formation, and mass transport [1]. Two dimensional convection-diffusion equation is given by the following equation. ( ) 1 ut + uux + uuy − R uxx + uyy = 0 (1). The Dirichlet boundary conditions are given by.

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