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

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.

Modeling Multiscale and Multiphysics Coastal Ocean Processes: A Discussion on Necessity, Status, and Advances

“…In a broader sense, domain decomposition is an indispensable approach to achieving multiscale and multiphysics simulations. A theoretical study on domain decomposition and data assimilation in the computation of a linearized version of the shallow water equations was demonstrated for Baltic Sea circulation [28,29]. In this study, assimilation data with randomness (to mimic observation data) was imposed at an open boundary of a subdomain that was linked to another subdomain.…”

“…In this study, assimilation data with randomness (to mimic observation data) was imposed at an open boundary of a subdomain that was linked to another subdomain. Its computation was formulated into an inverse problem, whose objective function was built on the governing equations and boundary conditions with a term of the Tikhonov regularization [28]. Additionally, a discussion on uniqueness and computational steps was presented.…”

“…Within ROMS, techniques are presented to realize two-way nesting to resolve local flows [60].In the past year, attempts have been made to analyze methods and computation, although such analyses are difficult and progress is limited due to the complexity of involved governing equations and also ocean flows. Research examples include optimal interface conditions to couple hydrostatic and nonhydrostatic ocean models [61], variational data assimilation [28], computational algorithms [62], and interface algorithms and stability analysis [63]. In a broad sense, all of these efforts fall into heterogeneous domain decomposition methods for coupling and computation with different partial differential equations, distinct numerical methods, and even dissimilar meshes [12].…”

“…Examples include numerical oscillations, delay of response in solution, and even non-physical solutions occurring at interfaces [64]. Domain decomposition is a broad framework for coupling of different models, and its research has been extensive in the mathematics community, e.g., [12,14], but with very sparse efforts for problems of interests to the ocean science community, e.g., references [27,28] collected in this Special Issue. Therefore, advances of domain decomposition methods are yet to be made directly for ocean flows.…”

Data analysis for variational assimilation of the surface temperature of the Black and Azov Seas

“…Other variants of using the considered algorithm in models of hydrothermodynamics of seas and oceans can be found for example in [Agoshkov et al, 2019;Sheloput, 2018].…”