×

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

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.

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
PDFBibTeX XMLCite
Full Text: EuDML EMIS