Convergence of Finite Difference Schemes for a Multidimensional Boussinesq Equation

Abstract

Abstract. Conservative finite difference schemes for the numerical solution of multi-dimensional Boussinesq-type equations are constructed and studied theoretically. Depending on the way the nonlinear term f (u) is approximated, two families of finite difference schemes are developed. Error estimates for these numerical methods in the uniform metric and the Sobolev space W 1 2 are obtained. The extensive numerical experiments given in [7] for the one-dimensional problem show good precision and full agreement between the theoretical results and practical evaluation for single soliton and the interaction between two solitons.

Invariant preserving schemes for double dispersion equations

“…Hence, we do not expect that the semidiscrete momentum (31) will be exactly preserved after applying a symplectic integration method to (8). However, we prove in Theorem 6 that the solution to the symplectic scheme FDS-S approximately preserves the discrete momentum (17) with O(h 2 ) global error.…”

“…We analyze numerically the preservation of the discrete energy, computed by Eq. (26), and the discrete momentum, evaluated by (17).…”

“…For consistency of presentation we also formulate the following well studied scheme (see e.g. [17,19]):…”

“…The finite difference method is used in [7,9,10,16,18] to solve the one-dimensional problem (1). The exact preservation of the global discrete energy and the convergence of the numerical methods are proved in [17][18][19].…”

Comparison of Some Finite Difference Schemes for Boussinesq Paradigm Equation

“…The properties of some particular cases of these three families were studied theoretically in our papers [8,9]. We have proved that all schemes considered above have second order of convergence in space and time.…”