Construction of a New Implicit Difference Scheme for 2D Boussinesq Paradigm Equation

Given an orthogonal and uniform solution grid with equal spatial grid sizes, we construct a new second-order implicit conservative finite difference scheme for the fourth-order 2D Boussinesq paradigm equation with quadratic nonlinear part. We apply the algebraic approach to the construction of difference schemes suggested by the first two authors and based on a combination of the finite volume method, difference elimination, and numerical integration. For the difference elimination, we make use of the techniques of Gröbner bases; in so doing, we introduce an extra difference indeterminate to reduce the nonlinear elimination problem to the pure linear one. It allows us to apply the Gröbner bases algorithm and software designed for linear generating sets of difference polynomials. Additionally, for the obtained difference scheme and also for another scheme known in the literature, we compute the modified differential equations and compare them.