Volume 31, Issue 1

Special semidiscrete approximations, namely, basic, first, and second semidiscrete problems are proposed for a boundary value problem describing a stationary radiative–conductive heat transfer in a two-dimensional system of heat-conductive plates of width ɛ separated by vacuum interlayers. We prove the unique solvability for the first and second semidiscrete problems, establish a comparison theorem, and obtain estimates of their solutions. For the first semidiscrete problem we also obtain some results concerning stability and derive an error estimate of order O ( ɛ ). Results of numerical experiments are presented.

Modern atmospheric models for climate simulations require accurate and efficient, locally mass-conservative and monotonic numerical schemes for treating the transport of atmospheric constituents. One of the ways to design such schemes is Finite-Volume Semi-Lagrangian approach (FVSL). FVSL schemes are characterised by the computational efficiency advantage due to the possibility of using large time-steps and efficient treatment of multiple transported quantities (tracers). This article presents massively parallel and multi-tracer efficient version of the recently developed 3D cascade FVSL transport scheme. Using hybrid distributed-shared memory parallelism with 1D MPI domain decomposition in latitude and OpenMP computations for longitude loops allows to use efficiently up to 1600 computational cores. We hope this number will grow with the future growth of the number of shared memory cores per computational node. Multi-tracer optimisations of the scheme (mostly, developing multi-tracer efficient monotonic filter) allow to reduce the cost of transporting additional tracer to 18–23% of running the scheme with one tracer.

In this paper we consider linear systems with dense-matrices which arise from numerical solution of boundary integral equations. Such matrices can be well-approximated with ℋ 2 -matrices. We propose several new preconditioners for such matrices that are based on the equivalent sparse extended form of ℋ 2 -matrices. In the numerical experiments we show that the most efficient approach is based on the so-called reverse-Schur preconditioning technique.

A discontinuous Galerkin method on unstructured grids is adapted and implemented for simulation of wave response of subvertical fractured systems in carbonate rocks for numerical solution of direct problems of seismic exploration. The present paper compares seismic responses for several mechanical-mathematical models of a fractured reservoir. The models of collectors differ in the presence and location of media interfaces relative to the collector and also in a way of its specification, namely, an explicit selection of fractures with the parameters of the medium in the domain of collector coinciding with the host medium, or differing from it. We indicate the ability to take into account inter-fracture interactions with the use of the model of a fractured layer presented in the paper and study wave processes formed as the result of interaction of seismic pulses with fractured reservoirs.

We consider the generalized time-dependent Schrödinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step h . Next, for a selected scheme of the family, we discover that the discrete convolution in time in the discrete TBC does not depend on h and, moreover, it coincides with the corresponding convolution in the semi-discrete TBC rewritten similarly. Both moments allow us to prove the bound for the difference between the kernels of the discrete convolutions in the discrete and semi-discrete TBCs that is the first result of such kind. Numerical experiments on replacing the discrete TBC convolutions by the semi-discrete one exhibit truly small absolute errors though not relative ones in general. The suitable discretization in space of the semi-discrete TBC for the higher-order Numerov scheme is also discussed.