A LINEAR TIME-SPACE HIGH-ORDER COMPACT FINITE DIFFERENCE SCHEME FOR SOLVING THE ONE-DIMENSIONAL NONLINEAR BURGERS' EQUATION

This paper presents a novel finite difference scheme for solving the one-dimensional (1D) nonlinear Burgers' equation with high accuracy. A fourth-order backward difference formula is employed to discretize the time derivative, while the nonlinear residual terms are efficiently linearized via Taylor series expansion. Spatial discretization is achieved using a fourth-order compact difference scheme for second-order derivatives and a fourth-order Padé scheme for first-order derivatives. As a result, the proposed approach yields a linear compact finite difference scheme with fourth-order accuracy in both temporal and spatial dimensions. The accuracy and stability of this method are demonstrated through a series of numerical experiments, validating its effectiveness for solving nonlinear Burgers' equations. The high-order finite difference scheme for the nonlinear Burgers' equation can accurately simulate phenomena in fields such as physics, sound wave propagation, and aerodynamics.