On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems With Nonlocal Boundary Conditions

Abstract

In this paper the one-and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized.

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.…”

“…The stability analysis of parabolic and pseudoparabolic problems with nonlocal boundary conditions is done by various methods. A short review of these methods is presented in [7].…”

“…One general technique to prove necessary and sufficient stability conditions for non-stationary numerical approximations of differential problems is to apply the eigenvalue criterion for non-normal matrices [7,13]. This stability analysis technique is applied for simple parabolic and pseudo-parabolic problems with nonlocal boundary conditions in [12,14].…”

“…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.…”

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

“…Two classes of integration methods are investigated. First, we use stability results obtained in [1], where unconditionally stable approximations of the pseudoparabolic problem were constructed only using the full approximation backward Euler method. Therefore this method is also used to construct parallel algorithms.…”