Linearly compact scheme for 2D Sobolev equation with Burgers’ type nonlinearity

In this paper, a bilinear three-point fourth-order compact operator is applied to solve the two-dimensional (2D) Sobolev equation with a Burgers’ type nonlinearity. In order to derive a high-order compact difference scheme, an effective reduction technique for the diffusion term is utilized to convert the original high-order evolutionary equation into a low-order system of equations equivalently, which could be discretized combining a classical linear compact operator and the bilinear compact operator. A three-level averaging technique in temporal dimension is embedded, which exports to a linearized compact difference scheme. The proposed compact difference scheme is proved to be uniquely solvable and convergent based on the energy analysis with the error estimate \(O(\tau ^{2}+{h_{1}^{4}}+{h_{2}^{4}})\) in L²-norm and H¹-norm, where h₁, h₂, and τ denote the spatial and temporal mesh step sizes, respectively. Extensive numerical experiments are carried out to validate the theoretical results on the convergence rates even though the dynamic capillary coefficient tends to zero. Further extending the idea to the 2D Benjamin-Bona-Mahony-Burgers (BBMB) equation is available. The accuracy of the proposed scheme is also demonstrated compared with that of a second-order numerical scheme without the compact technique.