Numerical computation of convection-diffusion-reaction equation using an improved explicit finite difference method

The convection-diffusion-reaction equation is one of the most useful equations in predicting many physical and chemical phenomena such as heat transfer, chemical species transport and reaction and the pollutants adsorption in wastewater. The widely used Upwind Forward Euler method may produce a negative solution because a time step size value is more than 0.005 and may form numerical oscillations. An improvement in the proper selection of time step size values is suggested, which removes the numerical oscillation and significantly improves the accuracy and convergence of the solution. In this paper, we add two explicit finite difference methods: Forward Time Central Space (FTCS) and Lax-Friedrichs, together with the previous method, Upwind, to solve the convection-diffusion-reaction equation with the improvement of time step size. An example is drawn from the literature to test the results of the numerical methods. The exact solution is used for comparison. Finally, the Upwind method solves the convection-diffusion-reaction problem satisfactorily with the restricted time step size, while the FTCS and Lax method form unstable solutions and thus causes the methods to be inapplicable for solving the convection-diffusion-reaction equation.