Volume 33, Issue 2

We formulate a class of inverse problems in the theory of ionosphere and study the methods of its solution based on variational data assimilation for ‘total electronic content’ (TEC). The issues of unique and dense solvability are considered and the solution algorithm for the problem formulated here is proposed.

In this paper, we propose and investigate a new homogenization method for diffusion problems in domains with multiple inclusions with large values of diffusion coefficients. The diffusion problem is approximated by the P1-finite element method on a triangular mesh. The underlying algebraic problem is replaced by a special system with a saddle point matrix. For the solution of the saddle point system we use the typical asymptotic expansion. We prove the error estimates and convergence of the expanded solutions. Numerical results confirm the theoretical conclusions.

The paper proposes a stable time-splitting method for the numerical simulation of free-surface viscous flows. The key features of the method are a semi-Lagrangian scheme for the level-set function transport improved with MacCormack predictor–corrector step with limiting strategy and an adaptive volume-correction procedure. The spatial discretization is done by a hybrid finite volume/finite difference method on dynamically adaptive hexahedral meshes. Numerical verification is done by comparing full-scale 3D numerical simulations of the sloshing tank and the coastal wave run-up with other numerical and experimental results known from the literature.

Dynamic processes in block-structured media are described within the framework of a mathematical model where elastic blocks interact with each other through pliable interlayers possessing complex mechanical properties. The case of elastic interaction is considered, the effects of plasticity and viscoelastic shear in block interlayers are taken into account. We use the model of porous fluid-saturated material in interlayers in which pores collapse during sudden or prolonged application of compressive stresses. An implicit difference scheme is constructed for numerical implementation of the model based on principles for construction of Ivanov’s schemes with controlled energy dissipation. Using MPI (Message Passing Interface), we developed parallel software for modelling wave motion in a two-dimensional formulation. The results of numerical analysis are presented for the waves initiated by rotation of blocks, which indicates a strong anisotropy of the blocky medium.

The paper demonstrates different ways of using the semi-Lagrangian approximation depending on the fulfillment of conservation laws. A one-dimensional continuity equation and a parabolic one are taken as simple methodological examples. For these equations, the principles of constructing discrete analogues are demonstrated for three different conservation laws (or the requirements of stability in the related discrete norms similar to the L 1 , L 2 , L ∞ -norms). It is significant that different conservation laws yield difference problems of different types as well as different ways to justify their stability.

The problem of variational data assimilation for an evolution model is considered with the aim to identify the initial condition. The solvability of the optimality system is studied. Based on the adjoint equations, iterative algorithms for solving the problem are developed and justified.