A parameter-uniform numerical method for a Sobolev problem with initial layer


AMIRALIYEV G. M. , Duru H. , AMIRALIYEVA I. G.

NUMERICAL ALGORITHMS, cilt.44, ss.185-203, 2007 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 44 Konu: 2
  • Basım Tarihi: 2007
  • Doi Numarası: 10.1007/s11075-007-9096-0
  • Dergi Adı: NUMERICAL ALGORITHMS
  • Sayfa Sayıları: ss.185-203

Özet

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.