Benchmark travelling waves and bidirectional adaptive DA–PMA computation for the two-dimensional regularized long-wave equation

This paper presents an adaptive computational framework for the two-dimensional regularized long-wave (RLW) equation. The main focus was on the numerical treatment of the model on moving nonuniform meshes. The RLW equation was first rewritten in a mixed differential-algebraic (DA) form by introducing an auxiliary variable. This reformulation avoids the direct discretization of the mixed space–time derivatives and leads to a more convenient adaptive implementation. The spatial approximation was constructed on bidirectionally adaptive tensor-product grids using five-point Fornberg finite-difference operators. Mesh redistribution was performed through a parabolic Monge–Ampère (PMA) rezoning strategy combined with monitor functions and shape-preserving interpolation between successive meshes. Exact travelling-wave solutions were also derived through an oblique wave reduction. These solutions were used mainly as benchmark data for the numerical computations. In particular, the solitary-wave solution provides compatible initial conditions, boundary data, auxiliary-field values, and reference profiles for error measurement. The stability discussion was carried out under admissible mesh evolution. On each frozen mesh, the spatial approximation was formally second-order consistent. The convergence behavior of the full adaptive algorithm was assessed numerically because mesh redistribution and inter-mesh interpolation introduce additional error effects. Numerical experiments were presented for single-wave and two-pulse configurations. The results show that the adaptive method preserves the wave structure with good accuracy and produces smaller errors than the corresponding fixed-mesh computation. The adaptive meshes remained concentrated near the dominant wave regions while maintaining global mesh regularity throughout the simulation.