MODIFIED FINITE-DIFFERENCE SCHEME FOR SOLVING ONE CLASS OF CONVECTION-REACTION-DIFFUSION PROBLEMS

The paper reviews approaches to the construction of finite-difference methods for solving time-dependent diffusion equations and transport equations. A modified computational scheme for solving a semilinear multidimensional equation of the «reaction – diffusion – convection» type is presented. The hybrid computational scheme is based on the alternating directions method and the Robert-Weiss scheme.

Numerical Study Of Thermal Behavior Of Anisotropic Medium In Bidirectional Cylindrical Geometry

In this work, we investigate the transient thermal analysis of two-dimensional cylindrical anisotropic medium subjected to a prescribed temperature at the two end sections and to a heat flux over the whole lateral surface. Due to the complexity of analytically solving the anisotropic heat conduction equation, a numerical solution has been developed. It is based on a coordinate transformation that reduces the anisotropic cylinder heat conduction problem to an equivalent isotropic one, without complicating the boundary conditions but with a more complicated geometry. The equation of heat conduction for this virtual medium is solved by the alternating directions method. The inverse transformation makes it possible to determine the thermal behavior of the anisotropic medium as a function of study parameters: diagonal and cross thermal conductivities, heat flux.

Accelerated method for solving grid equations II (3 – D case)

In this paper following the parametric extrapolation technique [1] we consider the case when the starting operator is split into the sum of three operators. A result on estimating iteration numbers needed for solving the perturbed problem is obtained. Finally, the advantage of our method over the direct use of the alternating directions method is shown on examples.

On the $O(1/k)$ Convergence Rate of He's Alternating Directions Method for a Kind of Structured Variational Inequality Problem

The alternating directions method for a kind of structured variational inequality problem (He, 2001) is an attractive method for structured monotone variational inequality problems. In each iteration, the subproblemsare convex quadratic minimization problem with simple constraintsand a well-conditioned system of nonlinear equations that can be efficiently solvedusing classical methods. Researchers have recently described the convergence rateof projection and contraction methods for variational inequality problems andthe original ADM and its linearized variant. Motivated and inspired by researchinto the convergence rate of these methods, we provide a simple proof to show the $O(1/k)$ convergencerate of alternating directions methods for structured monotone variational inequality problems (He, 2001).

A Line-Search-Based Partial Proximal Alternating Directions Method for Separable Convex Optimization

We propose an appealing line-search-based partial proximal alternating directions (LSPPAD) method for solving a class of separable convex optimization problems. These problems under consideration are common in practice. The proposed method solves two subproblems at each iteration: one is solved by a proximal point method, while the proximal term is absent from the other. Both subproblems admit inexact solutions. A line search technique is used to guarantee the convergence. The convergence of the LSPPAD method is established under some suitable conditions. The advantage of the proposed method is that it provides the tractability of the subproblem in which the proximal term is absent. Numerical tests show that the LSPPAD method has better performance compared with the existing alternating projection based prediction-correction (APBPC) method if both are employed to solve the described problem.

IMPROVED ACCURACY FOR ALTERNATING-DIRECTION METHODS FOR PARABOLIC EQUATIONS BASED ON REGULAR AND MIXED FINITE ELEMENTS

An efficient modification by Douglas and Kim of the usual alternating directions method reduces the splitting error from [Formula: see text] to [Formula: see text] in time step k. We prove convergence of this modified alternating directions procedure, for the usual non-mixed Galerkin finite element and finite difference cases, under the restriction that k/h2 is sufficiently small, where h is the grid spacing. This improves the results of Douglas and Gunn, who require k/h4 to be sufficiently small, and Douglas and Kim, who require that the locally one-dimensional operators commute. We propose a similar and efficient modification of alternating directions for mixed finite element methods that reduces the splitting error to [Formula: see text], and we prove convergence in the noncommuting case, provided that k/h2 is sufficiently small. Numerical computations illustrating the mixed finite element results are also presented. They show that our proposed modification can lead to a significant reduction in the alternating direction splitting error.