Applications of High-Order Compact Finite Difference Scheme to Nonlinear Goursat Problems
Several numerical schemes utilizing central difference
approximations have been developed to solve the Goursat problem.
However, in a recent years compact discretization methods which
leads to high-order finite difference schemes have been used since it
is capable of achieving better accuracy as well as preserving certain
features of the equation e.g. linearity. The basic idea of the new
scheme is to find the compact approximations to the derivative terms
by differentiating centrally the governing equations. Our primary
interest is to study the performance of the new scheme when applied
to two Goursat partial differential equations against the traditional
finite difference scheme.
[1] J. Zhao, T. Zhang and R.M. Corless. Convergence of the compact finite
difference method for second order elliptic equations. Applied
Mathematics and Computation. 2006, pg. 1454-1469.
[2] J. Li and M.R. Visbal. High order compact schemes for nonlinear
dispersive waves. Journal of scientific computing. 2006, pg. 1-23.
[3] B.J. Boersma. A staggered compact finite difference formulation for the
compressible Navier-Stokes equations. Journal of Computational
Physics. 2005, pg. 675-690.
[4] J. Zhao, R.M. Corless and M. Davison. Financial applications of
symbolically generated compact finite difference formulae. Dept. of
Applied Mathematics. Western Ontario London University. 2005.
[5] J. Li and Y. Chen. High order compact schemes for dispersive media.
Electronics Letters. 2004, pg. 14.
[6] Y. Kyei. Higher order Cartesian grid based finite difference methods for
elliptic equations on irregular domains and interface problems and their
applications. PhD dissertation. North Carolina State University. 2004.
[7] M.A.S. Nasir and A.I.M. Ismail. Numerical solution of a linear Goursat
problem: Stability, consistency and convergence. WSEAS Transactions
on Mathematics. 2004.
[8] M. Li and T. Tang. A compact fourth order finite difference scheme for
unsteady viscous incompressible flows. Journal of Scientific Computing.
2001, pg. 29-45.
[9] M.O. Ahmed. An exploration of compact finite difference methods for
the numerical solution of PDE. PhD dissertation. Western Ontario
London University. 1997.
[10] R. Bodenmann and H.J. Schroll. Compact difference methods applied to
initial boundary value problems for mixed systems. Numer. Math. 1996,
pg. 291-309.
[11] A.M. Wazwaz. The decomposition method for approximate solution of
the Goursat problem. Applied Mathematics and Computation. 1995, pg.
299-311.
[12] W.F. Spotz. High order compact finite difference schemes for
computational mechanics. PhD dissertation. Texas University. 1995.
[13] J.R. Mclaughlin, P.L. Polyakov and P.E. Sacks. Reconstruction of a
spherical symmetric speed of sound. SIAM Journal of Applied
Mathematics. 1994, pg. 1203-1223.
[14] A.M. Wazwaz. On the numerical solution for the Goursat problem.
Applied Mathematics and Computation. 1993, pg. 89-95.
[15] P. Hillion. A note on the derivation of paraxial equation in
nonhomogeneous media. SIAM Journal of Applied Mathematics. 1992,
pg. 337-346.
[16] D.J. Evans and B.B. Sanugi. Numerical solution of the Goursat problem
by a nonlinear trapezoidal formula. Applied Mathematics Letter. 1988,
pg. 221-223.
[17] E.H. Twizell. Computational Methods For Partial Differential
Equations. Ellis Horwood Limited. 1984.
[18] D.J. Kaup and A.C. Newell. The Goursat and Cauchy problems for the
Sine Gordon equation. SIAM Journal of Applied Mathematics. 1978, pg.
37-54.
[19] T.Y. Cheung. Three Nonlinear Initial Value Problems of The Hyperbolic
Type. SIAM J. Numer. Anal. 1977, pg. 484-491.
[20] R.A. Frisch and B.R. Cheo. On a bounded one dimensional Poisson-
Vlasov system. Society for Industry and Applied Mathematics. 1972.
[21] J.T. Day. A Runge-Kutta method for the numerical solution of the
Goursat problem in hyperbolic partial differential equations. Computer
Journal. 1966, pg. 81-83.
[1] J. Zhao, T. Zhang and R.M. Corless. Convergence of the compact finite
difference method for second order elliptic equations. Applied
Mathematics and Computation. 2006, pg. 1454-1469.
[2] J. Li and M.R. Visbal. High order compact schemes for nonlinear
dispersive waves. Journal of scientific computing. 2006, pg. 1-23.
[3] B.J. Boersma. A staggered compact finite difference formulation for the
compressible Navier-Stokes equations. Journal of Computational
Physics. 2005, pg. 675-690.
[4] J. Zhao, R.M. Corless and M. Davison. Financial applications of
symbolically generated compact finite difference formulae. Dept. of
Applied Mathematics. Western Ontario London University. 2005.
[5] J. Li and Y. Chen. High order compact schemes for dispersive media.
Electronics Letters. 2004, pg. 14.
[6] Y. Kyei. Higher order Cartesian grid based finite difference methods for
elliptic equations on irregular domains and interface problems and their
applications. PhD dissertation. North Carolina State University. 2004.
[7] M.A.S. Nasir and A.I.M. Ismail. Numerical solution of a linear Goursat
problem: Stability, consistency and convergence. WSEAS Transactions
on Mathematics. 2004.
[8] M. Li and T. Tang. A compact fourth order finite difference scheme for
unsteady viscous incompressible flows. Journal of Scientific Computing.
2001, pg. 29-45.
[9] M.O. Ahmed. An exploration of compact finite difference methods for
the numerical solution of PDE. PhD dissertation. Western Ontario
London University. 1997.
[10] R. Bodenmann and H.J. Schroll. Compact difference methods applied to
initial boundary value problems for mixed systems. Numer. Math. 1996,
pg. 291-309.
[11] A.M. Wazwaz. The decomposition method for approximate solution of
the Goursat problem. Applied Mathematics and Computation. 1995, pg.
299-311.
[12] W.F. Spotz. High order compact finite difference schemes for
computational mechanics. PhD dissertation. Texas University. 1995.
[13] J.R. Mclaughlin, P.L. Polyakov and P.E. Sacks. Reconstruction of a
spherical symmetric speed of sound. SIAM Journal of Applied
Mathematics. 1994, pg. 1203-1223.
[14] A.M. Wazwaz. On the numerical solution for the Goursat problem.
Applied Mathematics and Computation. 1993, pg. 89-95.
[15] P. Hillion. A note on the derivation of paraxial equation in
nonhomogeneous media. SIAM Journal of Applied Mathematics. 1992,
pg. 337-346.
[16] D.J. Evans and B.B. Sanugi. Numerical solution of the Goursat problem
by a nonlinear trapezoidal formula. Applied Mathematics Letter. 1988,
pg. 221-223.
[17] E.H. Twizell. Computational Methods For Partial Differential
Equations. Ellis Horwood Limited. 1984.
[18] D.J. Kaup and A.C. Newell. The Goursat and Cauchy problems for the
Sine Gordon equation. SIAM Journal of Applied Mathematics. 1978, pg.
37-54.
[19] T.Y. Cheung. Three Nonlinear Initial Value Problems of The Hyperbolic
Type. SIAM J. Numer. Anal. 1977, pg. 484-491.
[20] R.A. Frisch and B.R. Cheo. On a bounded one dimensional Poisson-
Vlasov system. Society for Industry and Applied Mathematics. 1972.
[21] J.T. Day. A Runge-Kutta method for the numerical solution of the
Goursat problem in hyperbolic partial differential equations. Computer
Journal. 1966, pg. 81-83.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:62257", author = "Mohd Agos Salim Nasir and Ahmad Izani Md. Ismail", title = "Applications of High-Order Compact Finite Difference Scheme to Nonlinear Goursat Problems", abstract = "Several numerical schemes utilizing central difference
approximations have been developed to solve the Goursat problem.
However, in a recent years compact discretization methods which
leads to high-order finite difference schemes have been used since it
is capable of achieving better accuracy as well as preserving certain
features of the equation e.g. linearity. The basic idea of the new
scheme is to find the compact approximations to the derivative terms
by differentiating centrally the governing equations. Our primary
interest is to study the performance of the new scheme when applied
to two Goursat partial differential equations against the traditional
finite difference scheme.", keywords = "Goursat problem, partial differential equation, finite
difference scheme, compact finite difference", volume = "6", number = "8", pages = "1156-5", }
