Low-Regularity Integrator and Inviscid Limit Problem for the KdV–Burgers Equation

This article investigates the inviscid limit and time discretizations for the Korteweg–de Vries–Burgers (KdV–B) equation on a torus, focusing on low-regularity solutions below [Formula: see text]. First, we establish an [Formula: see text] error estimate between the exact solutions of the KdV–Burgers and the KdV equations, achieving a convergence rate of order [Formula: see text]. Second, for numerical approximations, we propose a scheme that is uniformly convergent with respect to the diffusion coefficient [Formula: see text]. Under the regularity assumption [Formula: see text] for [Formula: see text], we prove that the proposed scheme achieves a convergence rate of order [Formula: see text] (up to a logarithmic factor) in [Formula: see text], with an error constant that is independent of [Formula: see text]. Consequently, an error bound is derived between the numerical solution of the KdV–Burgers equation and that of the standard KdV equation. Extensive numerical experiments, including comparisons of the proposed method with classical schemes, validate our theoretical results.