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**Analysis of Thacker’s method for solving the linearized shallow water equations.**
*(English)*
Zbl 0626.76016

Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 295-302 (1986).

[For the entire collection see Zbl 0595.00009.]

The differential operator L for the shallow water equations is suitably defined and noted that it is skew-adjoint and therefore, concluded from the theory of semigroups that the problem possesses a unique solution with certain properties. For a fixed triangularization of the domain, a space semi-discretization of the problem is introduced in the form Dẇ(t)\(=Aw(t)\), \(w(0)=w_ 0\). Since A is not antisymmetric as the operator L, the equations are modified so that the corresponding matrix becomes antisymmetric. Finally the time-discretization reduces to an algebraic problem. The stability and consistency properties are examined and the error estimates are obtained. The corresponding results of the original equations are then obtained from those of the modified equations.

The differential operator L for the shallow water equations is suitably defined and noted that it is skew-adjoint and therefore, concluded from the theory of semigroups that the problem possesses a unique solution with certain properties. For a fixed triangularization of the domain, a space semi-discretization of the problem is introduced in the form Dẇ(t)\(=Aw(t)\), \(w(0)=w_ 0\). Since A is not antisymmetric as the operator L, the equations are modified so that the corresponding matrix becomes antisymmetric. Finally the time-discretization reduces to an algebraic problem. The stability and consistency properties are examined and the error estimates are obtained. The corresponding results of the original equations are then obtained from those of the modified equations.

Reviewer: V.Subba Rao

### MSC:

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

35Q99 | Partial differential equations of mathematical physics and other areas of application |