Volume 34, Issue 1

We propose a numerical algorithm for simulations of two-component viscous compressible isothermal flows with surface effects in 3D domains of complex shape with voxel representation of geometry. The basic mathematical model is the regularized system of Navier–Stokes–Cahn–Hilliard equations. Simulations of droplet spreading over a flat base and displacement of one liquid by another one in a pore space of real rock sample are carried out. The simulation results demonstrate the applicability and good efficiency of the used system of equations, the corresponding difference scheme, and its implementation algorithms for numerical solution of the considered class of problems.

A new approach to formulation of asymptotic boundary conditions for eigenvalue problems arising in numerical analysis of hydrodynamic stability of such shear flows as boundary layers, separations, jets, wakes, characterized by almost constant velocity of the main flow outside the shear layer or layers is proposed and justified. This approach makes it possible to formulate and solve completely the temporal and spatial stability problems in the locally-parallel approximation, reducing them to ordinary algebraic eigenvalue problems.

The adaptive market hypothesis (AMH) supplies a convincing motivation for why market efficiency should not be regarded as a stable property in time. This paper explores a Bayesian methodology for estimating weak-form market efficiency under the AMH using a test of evolving efficiency (TEE). More precisely, a generalized TEE (GTEE) approach is proposed in which the conditional first moment of a time series is assumed to be a nonlinear function of its conditional second moment, i.e., a nonlinear feedback term is present in the conditional mean equation. We then discuss a maximum likelihood estimation procedure for the resulting nonlinear model using the state-space approach and extended Kalman filtering. This methodology is used to estimate time-varying, weak-form market efficiency in four, specifically chosen, markets over a time-period that includes the global financial crisis of 2007/2008.

The weighted method of dependent trials or weighted method of correlated sampling (MCS) allows one to construct estimators for functionals based on the same Markov chain simultaneously for a given range of the problem parameters. Choosing an appropriate Markov chain, it is necessary to take into account additional conditions providing the finiteness of the computational cost of weighted MCS. In this paper we study the issue of finite computational cost of the method of correlated sampling (MCS) in application to evaluation of linear functionals of solutions to a set of systems of 2nd kind integral equations. A universal modification of the vector weighted MCS is constructed providing the branching of chain trajectory according to elements of matrix weights. It is proved that the computational cost of the constructed algorithm is bounded in the case the base functionals are also bounded. The results of numerical experiments using the modified weighted estimator are presented for some problems of the theory of radiation transfer subject to polarization.

Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material). Using a finite elements discretization (FEM) of the PDE and functional we obtain an integer programming problem. Due to approximation error of the FEM discretization, optimization problem becomes mesh-depended and possess false, physically inadequate optimums, while functional value heavily depends on the fineness of discretization scheme used to compute it. To alleviate this problem, we propose regularization of given functional by error estimate of the FEM discretization. This regularization provides robustness of solutions and improves obtained functional values as well. While the idea is broadly applicable, in this paper we apply our method to the heat conduction optimization. Problems of this type are of practical importance in design of heat conduction channels, heat sinks and other types of heat guides.