Numerical technique for solving convective-reaction-diffusion equation

By using splitting-up technique and cubic splines, a numerical algorithm of second order accuracy is developed to solve the nonlinear equation ut = Re−1uxx + [/tf(u)]x + h(u), with prescribed initial and boundary conditions. The validity of the difference scheme is tested by applying it to the nonlinear Burgersʹ equation. The numerical results obtained for Burgersʹ equation at low as well as for high Reynolds numbers are found to be in good agreement with the exact solutions. Then, the proposed scheme is used for solving the convective-reaction-diffusion equation ut = Re−1 uxx + ϵuux + αup, with initial and boundary conditions. Graphs have been drawn for the numerical solutions at low as well as for high Reynolds numbers by taking the values of parameters in 0 ≤ α ≤ 10, 0 < p ≤ 2, and 0 ≤ ϵ ≤ 2. The numerical solutions at low values of Re are in steady state for some sets of values of parameters. For intermediate values of Re, the solutions get affected considerably by convective and/or reactive forces, and for high values of Re the solutions blow up. The computed results are compared with the available results for some particular cases.