Parallel algorithms for three-dimensional parabolic and pseudoparabolic problems with different boundary conditions

Abstract

Abstract. In this paper, three-dimensional parabolic and pseudo-parabolic equations with classical, periodic and nonlocal boundary conditions are approximated by the full approximation backward Euler method, locally one dimensional and Douglas ADI splitting schemes. The stability with respect to initial conditions is investigated. We note that the stability of the proposed numerical algorithms can be proved only if the matrix of discrete operator can be diagonalized and eigenvectors make a complete basis system. Parallel versions of all algorithms are constructed and scalability analysis is done. It is shown that discrete one-dimensional problems with periodic and nonlocal boundary conditions can be efficiently solved with similar modifications of the parallel Wang algorithm.

On Stability Analysis of Finite Difference Schemes for Generalized Kuramoto-Tsuzuki Equation With Nonlocal Boundary Conditions

“…Here we apply the same stability analysis template as in [5,7]. The main idea is to split the stability analysis of the given differential equation (or time integration method) and the analysis of a structure of the spectrum for a specified discrete operator with various nonlocal boundary conditions.…”

“…In order to solve the obtained systems of linear equations with a matrix of special structure, we have used a superposition method to find a solution in the form [5,6]:…”

“…In papers [5,7], we have proposed a methodology, when the stability of the specified model is investigated in two steps, and the influence of the differential equation (and a finite-difference or finite-volume scheme) is separated from the analysis of the influence of nonlocal boundary conditions. Thus a big part of existing stability results can be reused if a new type of mathematical model with nonlocal boundary conditions is analysed.…”

Semi-Implicit Difference Scheme for a Two-Dimensional Parabolic Equation With an Integral Boundary Condition

“…In a two-dimensional case one of the first articles in this direction was [5]. Later on, numerical methods for such a problem were analyzed in the articles [1,7,15,18,22]. Particularly, the two-dimensional problems with nonlocal conditions using one-dimensional difference schemes were investigated in [6,10,20,25,31,32].…”

M-Matrices and Convergence of Finite Difference Scheme for Parabolic Equation With Integral Boundary Condition

“…In a two-dimensional case of parabolic equations with integral conditions, the difference methods were applied in [4,6,21,24,29], and to the equations with other type of nonlocal conditions -in [13,15].…”