A Structure-Preserving Covering Method for the KdV-Burgers Equation with Exact Conservation and High-Order Compact Discretization

Structure-preserving numerical methods are well-established for purely conservative or purely dissipative systems but remain underdeveloped for mixed-type equations coupling dispersion, dissipation, and nonlinearity. We investigate the Korteweg–de Vries–Burgers equation as a canonical model of this class. We develop a geometric covering method based on nonlocal symmetries that lifts the equation to an extended manifold, enabling exact conservation law preservation. As a pedagogical counterexample, we also analyze a naive recursive approximation. Both methods are implemented using sixth-order compact finite differences and fourth-order Runge–Kutta (RK4) time integration. Numerical experiments on sinusoidal waves, two-soliton collisions, and perturbed traveling waves show that the covering method reduces numerical dissipation by 50% and phase error by 90% relative to a standard second-order scheme, achieving one to two orders of magnitude higher accuracy. Mass and momentum are conserved to machine precision (below ${10}^{-14}$), and soliton amplitudes are preserved to within 0.3% after collision, with only 15% computational overhead. The framework offers a generalizable template for embedding nonlocal symmetries into high-order numerical methods for nonlinear wave equations.