Local discontinuous Galerkin finite element method for the nonlinear Korteweg–de Vries-Benjamin-Bona-Mahony-Burgers equation

In this work, the local discontinuous Galerkin (LDG) method is implemented to numerically solve the nonlinear Korteweg–de Vries-Benjamin-Bona-Mahony-Burgers equation, where an explicit four-stage third-order strong-stability-preserving Runge–Kutta scheme is applied for the temporal direction, and the LDG method is developed to approximate the spatial direction. The L2 stability of the LDG scheme is established, and a detailed a priori error estimate in the L2-The norm is derived by carefully choosing the interface numerical fluxes. Also, the exp(−Φ(ξ))-Expansion and Kudryashov techniques are applied in this work to derive novel exact traveling wave solutions that include solutions involving hyperbolic and trigonometric functions with arbitrary parameters. Finally, some numerical examples validating the analytical findings are presented with several two-dimensional and 3D graphs, as well as tabular data, which includes the L2 and L∞ error norms. The simulation results obtained for both P2 and P3 elements demonstrate the noteworthy accuracy and numerical proficiency of the LDG method in solving the governing nonlinear model.