Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis-Procesi equation

In this paper, the Degasperis-Procesi (DP) equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form: u - P . Based on the equivalent formulation of DP equation, a new numerical method is presented. The multi-quadric (MQ) quasi-interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order TVD method is used to approximate the time derivative. Several examples are presented to demonstrate the effectiveness of the proposed method.

[1]  Yinhua Xia,et al.  Fourier Spectral Methods for Degasperis–Procesi Equation with Discontinuous Solutions , 2014, J. Sci. Comput..

[2]  Giuseppe Maria Coclite,et al.  Numerical schemes for computing discontinuous solutions of the Degasperis–Procesi equation , 2007 .

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

[4]  Nianyu Yi,et al.  A conservative discontinuous Galerkin method for the Degasperis-Procesi equation , 2014 .

[5]  Zongmin Wu,et al.  Shape preserving properties and convergence of univariate multiquadric quasi-interpolation , 1994 .

[6]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[7]  Y. C. Hon,et al.  A quasi-radial basis functions method for American options pricing , 2002 .

[8]  M. J. D. Powell,et al.  Univariate multiquadric approximation: Quasi-interpolation to scattered data , 1992 .

[9]  Y. Liu,et al.  An operator splitting method for the Degasperis-Procesi equation , 2009, J. Comput. Phys..

[10]  Tony W. H. Sheu,et al.  A dispersively accurate compact finite difference method for the Degasperis-Procesi equation , 2013, J. Comput. Phys..

[11]  Yuto Miyatake,et al.  Conservative finite difference schemes for the Degasperis-Procesi equation , 2012, J. Comput. Appl. Math..

[12]  Ronghua Chen,et al.  Solving partial differential equation by using multiquadric quasi-interpolation , 2007, Appl. Math. Comput..

[13]  Zhiwu Lin,et al.  Stability of peakons for the Degasperis‐Procesi equation , 2007, 0712.2007.

[14]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[15]  Yongzhong Song,et al.  Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations , 2014 .

[16]  Limin Ma,et al.  Approximation to the k-th derivatives by multiquadric quasi-interpolation method , 2009, J. Comput. Appl. Math..

[17]  Yong Duan,et al.  A numerical scheme for nonlinear Schrödinger equation by MQ quasi-interpolation , 2013 .

[18]  Chi-Wang Shu,et al.  Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation , 2011 .

[19]  Lin-Tian Luh,et al.  The Mystery of the Shape Parameter II , 2010, 1002.2082.

[20]  Darryl D. Holm,et al.  A New Integrable Equation with Peakon Solutions , 2002, nlin/0205023.

[21]  Wenwu Gao,et al.  Solving time-dependent differential equations by multiquadric trigonometric quasi-interpolation , 2015, Appl. Math. Comput..

[22]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[23]  Zongmin Wu,et al.  Solving hyperbolic conservation laws using multiquadric quasi‐interpolation , 2006 .

[24]  W. Strauss,et al.  Stability of peakons , 2000 .

[25]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

[26]  Shengliang Zhang,et al.  Conservative multiquadric quasi-interpolation method for Hamiltonian wave equations , 2013 .