A linear finite difference scheme with error analysis designed to preserve the structure of the 2D Boussinesq paradigm equation

Use of the finite difference method has produced successful solutions to the general partial differential equations due to its efficiency and effectiveness with wide applications. For example, the 2D Boussinesq paradigm equation can be numerically studied using a linear‐implicit finite difference scheme based on the Crank‐Nicolson/Adams‐Bashforth technique. First, conservative quantities are derived and preserved through numerical scheme. Then, the convergence and stability analysis is then provided to simulate a numerical solution whose existence and uniqueness are proved based on the boundedness of the numerical solution. Analysis of spatial accuracy is found to be second order on a uniform grid. Numerical results from simulations indicate that these proposed scheme provide satisfactory second‐order accuracy both in time and space with an ‐norm, and also preserve discrete invariants. Additionally, previous scientific literature review has provided little evidence of studied terms with dispersive effect in 2D Boussinesq paradigm equation. The current study explores solution behavior by applying the proposed scheme to numerically analyze initial Gaussian condition.