A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations

In this paper, radial basis function (RBFs) based mesh-free method is implemented to find numerical solution of the Kuramoto-Sivashinsky equations. This approach has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. The accuracy of the method is assessed in terms of the error norms L"2,L"~, number of nodes in the domain of influence, free parameter, dependent parameter RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations.

[1]  W. Madych,et al.  Multivariate interpolation and condi-tionally positive definite functions , 1988 .

[2]  Kannan M. Moudgalya,et al.  Second-order splitting combined with orthogonal cubic spline collocation method for the Kuramoto-Sivashinsky equation , 1998 .

[3]  R. Grimshaw,et al.  Travelling wave solutions of the Kuramoto-Sivashinsky equation , 1988 .

[4]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[5]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[6]  Georgios Akrivis,et al.  Implicit-explicit BDF methods for the Kuramoto-Sivashinsky equation , 2004 .

[7]  Tian-Shiang Yang.  On traveling-wave solutions of the Kuramoto-Sivashinsky equation , 1997 .

[8]  Benny Y. C. Hon,et al.  An efficient numerical scheme for Burgers' equation , 1998, Appl. Math. Comput..

[9]  I. Dag,et al.  Numerical solutions of KdV equation using radial basis functions , 2008 .

[10]  M. A. López-Marcos.  Numerical analysis of pseudospectral methods for the Kuramoto-Sivashinsky equation , 1994 .

[11]  R. Franke.  A Critical Comparison of Some Methods for Interpolation of Scattered Data , 1979 .

[12]  S. Zaki,et al.  A least-squares finite element scheme for the EW equation , 2000 .

[13]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[14]  A. P. Hooper,et al.  Nonlinear instability at the interface between two viscous fluids , 1985 .

[15]  G. Sivashinsky,et al.  Instabilities, Pattern Formation, and Turbulence in Flames , 1983 .

[16]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[17]  W. Holter,et al.  A vectorized multigrid solver for the three-dimensional Poisson equation , 1986 .

[18]  J. Hyman,et al.  The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems , 1986 .

[19]  D Michelson,et al.  Steady solutions of the Kuramoto-Sivashinsky equation , 1986 .

[20]  Y. Kuramoto,et al.  Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium , 1976 .

[21]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[22]  Carsten Franke,et al.  Convergence order estimates of meshless collocation methods using radial basis functions , 1998, Adv. Comput. Math..

[23]  R. E. Carlson,et al.  The parameter R2 in multiquadric interpolation , 1991 .

[24]  Jaime Peraire,et al.  A time-adaptive finite volume method for the Cahn-Hilliard and Kuramoto-Sivashinsky equations , 2008, J. Comput. Phys..

[25]  Yan Xu,et al.  Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations , 2006 .

[26]  Eitan Tadmor,et al.  The well-posedness of the Kuramoto-Sivashinsky equation , 1986 .