*Article* **Domain Decomposition Method for the Variational Assimilation of the Sea Level in a Model of Open Water Areas Hydrodynamics**

#### **Valery Agoshkov 1,2, Natalia Lezina 1,\* and Tatiana Sheloput 1**


Received: 30 April 2019; Accepted: 20 June 2019; Published: 23 June 2019

**Abstract:** One of the modern fields in mathematical modelling of water areas is developing hybrid coastal ocean models based on domain decomposition. In coastal ocean modelling a problem to be solved is setting open boundary conditions. One of the methods dealing with open boundaries is variational data assimilation. The purpose of this work is to apply the domain decomposition method to the variational data assimilation problem. The method to solve the problem of restoring boundary functions at the liquid boundaries for a system of linearized shallow water equations is studied. The problem of determining additional unknowns is considered as an inverse problem and solved using well-known approaches. The methodology based on the theory of optimal control and adjoint equations is used. In the paper the theoretical study of the problem is carried out, unique and dense solvability of the problem is proved, an iterative algorithm is proposed and its convergence is studied. The results of the numerical experiments are presented and discussed.

**Keywords:** open boundaries; domain decomposition; variational data assimilation; inverse problems; shallow water equations; boundary conditions; mathematical modelling; coastal ocean modelling; computational methods
