Among inverse problems for partial differential equations, a task of interest is to study coefficient inverse problems related to identifying the right-hand side of an equation with the use of additional information. In the case of nonstationary problems, finding the dependence of the right-hand side on time and the dependence of the right-hand side on spatial variables can be treated as independent tasks. These inverse problems are linear, which considerably simplifies their study. The time dependence of the right-hand side of a multidimensional parabolic equation is determined using an additional solution value at a point of the computational domain. The inverse problem for a model equation in a rectangle is solved numerically using standard spatial difference approximations. The numerical algorithm relies on a special decomposition of the solution whereby the transition to a new time level is implemented by solving two standard grid elliptic problems. Numerical results are presented.

In this work, we consider the heat transfer problems with phase change. The mathematical model is described through a two-phase Stefan problem and defined in the whole domain that contains frozen and thawed subdomains. For the numerical solution of the problem, we present three schemes based on different smoothing of the sharp phase change interface. We propose the method using smooth coefficient approximation based on the analytical smoothing of discontinuous coefficients through an error function with a given smoothing interval. The second method is based on smoothing in one spatial interval (cell) and provides a minimal length of smoothing calculated automatically for the given values of temperatures on the mesh. The third scheme is a convenient scheme using a linear approximation of the coefficient on the smoothing interval. The results of the numerical computations on a model problem with an exact solution are presented for the one-dimensional formulation. The extension of the method is presented for the solution of the two-dimensional problem with numerical results.

In this work we present results of numerical simulation of three-dimensional temperature fields in soils for various applied problems: the railway line in the conditions of permafrost for different geometries, the horizontal tunnel underground storage and greenhouses of various designs in the Far North. Mathematical model of the process is described by a nonstationary heat equation with phase transitions of pore water. The numerical realization of the problem is based on the finite element method using a library of scientific computing FEniCS. For numerical calculations we use high-performance computing systems.

Under consideration are the numerical methods for simulation of a fluid flow in fractured porous media. The fractures are taken into account explicitly by using a discrete fracture model. The...