The numerical solutions for the nonhomogeneous Burgers' equation with the generalized Hopf-Cole transformation

In this paper, with the help of the generalized Hopf-Cole transformation, we first convert the nonhomogeneous Burgers' equation into an equivalent heat equation with the derivative boundary conditions, in which Neumann boundary conditions and Robin boundary conditions can be viewed as its special cases. For easy derivation and numerical analysis, the reduction order method is used to convert the problem into an equivalent first-order coupled system. Next, we establish a box scheme for this first-order system. By the technical energy analysis method, we obtain the prior estimate of the numerical solution for the box scheme. Furthermore, the solvability and convergence are obtained directly from the prior estimate. The extensive numerical examples are carried out, which verify the developed box scheme can achieve global second-order accuracy for both homogeneous and nonhomogeneous Burgers' equations.