On the convergence of a three-level vector SOR scheme. (English) Zbl 0888.65106

Jovanović, B. S.

Mat. Vesn. 47, No. 3-4, 93-98 (1995).

The first initial-boundary value problem for the multidimensional wave equation \[ \frac{\partial^2u}{\partial t^2} = \Delta u + f, \qquad (x, t) \in \Omega \times (0,T) = (0,1)^n \times (0,T), \]
\[ u(x,0) = u_0(x), \qquad \frac{\partial u(x,0)}{\partial t} = u_1(x), \qquad x\in \Omega, \]
\[ u(x,t)=0, \qquad x \in \partial \Omega, \qquad t\in (0,T) \] is solved, under the assumption that the generalized solution belongs to the Sobolev space \(W_2^s(Q), s \geq 2.\)
An alternating directions finite difference scheme from the article by V. N. Abrashin [Differ. Uravn. 26, No. 2, 314-323 (1990; Zbl 0698.65063)] is used and a vector variant of the successive overrelaxation method is obtained. The stability and the convergence of this scheme are proved.

Reviewer: Ljiljana Cvetković (Beograd)

MSC:

65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L05	Wave equation
65F10	Iterative numerical methods for linear systems

Keywords:

multidimensional wave equation; alternating direction finite difference scheme; successive overrelaxation; stability; convergence

Citations:

Zbl 0698.65063; Zbl 0712.65089