High accuracy two-level implicit compact difference scheme for 1D unsteady biharmonic problem of first kind: application to the generalized Kuramoto–Sivashinsky equation

ABSTRACT

We propose a two-level implicit high order compact scheme for the one-dimensional unsteady biharmonic problem of first kind. The values of φ and ${\phi }_x$ are prescribed on the boundary. Using a combination of values of φ and ${\phi }_x$ at each grid point, the difference formula is derived for the unsteady biharmonic equation without discretizing the boundary conditions. The proposed method has second order time accuracy and fourth order space accuracy using just three grid points of a single compact stencil at every time level. The first order space derivative is also computed with same accuracy as a by-product of the method. Using the Von-Neumann analysis, the derived scheme is shown to be unconditionally stable. With a slight modification, the proposed method is applicable to solve singular problems. The performance of the proposed scheme is illustrated by numerical experiments done on a collection of test problems having physical significance including the nonlinear Kuramoto–Sivashinsky equation.

KEYWORDS:

• Compact difference scheme
• generalized Kuramoto–Sivashinsky equation
• extended Fisher–Kolmogorov equation
• block tri-diagonal matrices
• unconditional stability
• singularity

AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M22

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

R.K. Mohanty  http://orcid.org/0000-0001-6832-1239