Volume 26, Issue 2
Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation

Tingchun Wang & Boling Guo

J. Part. Diff. Eq., 26 (2013), pp. 107-121.

Published online: 2013-06

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  • Abstract

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is also decoupled in computation, whichmeans that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in implementation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L^∞- norm, and for N in the discrete L^2-norm, respectively, where U and N are the numerical solutions of the KGZ equation. Numerical results verify the theoretical analysis.

  • Keywords

Klein-Gordon-Zakharov equation decoupled and linearized difference scheme energy conservation solvability convergence

  • AMS Subject Headings

65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

iapcmwang@gmail.com (Tingchun Wang)

gbl@iapcm.ac.cn (Boling Guo)

  • BibTex
  • RIS
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@Article{JPDE-26-107, author = {Wang , Tingchun and Guo , Boling}, title = {Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation}, journal = {Journal of Partial Differential Equations}, year = {2013}, volume = {26}, number = {2}, pages = {107--121}, abstract = {

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is also decoupled in computation, whichmeans that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in implementation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L^∞- norm, and for N in the discrete L^2-norm, respectively, where U and N are the numerical solutions of the KGZ equation. Numerical results verify the theoretical analysis.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v26.n2.2}, url = {http://global-sci.org/intro/article_detail/jpde/5156.html} }
TY - JOUR T1 - Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation AU - Wang , Tingchun AU - Guo , Boling JO - Journal of Partial Differential Equations VL - 2 SP - 107 EP - 121 PY - 2013 DA - 2013/06 SN - 26 DO - http://doi.org/10.4208/jpde.v26.n2.2 UR - https://global-sci.org/intro/article_detail/jpde/5156.html KW - Klein-Gordon-Zakharov equation KW - decoupled and linearized difference scheme KW - energy conservation KW - solvability KW - convergence AB -

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is also decoupled in computation, whichmeans that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in implementation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L^∞- norm, and for N in the discrete L^2-norm, respectively, where U and N are the numerical solutions of the KGZ equation. Numerical results verify the theoretical analysis.

Tingchun Wang & Boling Guo. (2019). Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation. Journal of Partial Differential Equations. 26 (2). 107-121. doi:10.4208/jpde.v26.n2.2
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