Numerical investigation of sixth-order BDF compact difference scheme for viscous Burgers' equations

This paper explores and analyses the sixth-order backward differentiation formula (BDF6) compact difference scheme for solving the viscous Burgers' equation, which is used to describe the nonlinear phenomena in flow. First, the time derivative is discretized by virtue of the BDF6 approach. Subsequently, we employ a compact difference method, combined with an order reduction technique, to carry out spatial discretization, thereby establishing the fully discrete scheme. Then, for the existence and uniqueness of solutions to numerical scheme, a detailed theoretical analysis are carried out. Furthermore, by using discrete energy argument and mathematical induction, combined with several essential lemmas related to the BDF6 multiplier, we derive the convergence and stability in a novel way. Finally, the accuracy and effectiveness of the proposed scheme is validated through several numerical examples.