Limits...
On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation.

Pan X, Zhang L - Springerplus (2016)

Bottom Line: In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation.The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time.The unique solvability of numerical solutions is shown.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematics and Information Science, Weifang University, Weifang, 261061 China ; School of Management, Qufu Normal University, Rizhao, 276800 China.

ABSTRACT
In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation. The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical "energy" of the system and finite difference method. The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time. The unique solvability of numerical solutions is shown. A priori estimate and fourth-order convergence of the finite difference approximate solution are discussed by using discrete energy method and some techniques of matrix theory. Numerical results are given to show the validity and the accuracy of the proposed method.

No MeSH data available.


Temporal convergence order in maximal norm for  at  with different h and  computed by the scheme (11)–(14)
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4835426&req=5

Fig2: Temporal convergence order in maximal norm for at with different h and computed by the scheme (11)–(14)

Mentions: In computations, we always choose the parameter . Take the parameters . To verify the accuracy in the spatial direction, we take . And we choose h small enough to verify the second-order accuracy in the temporal direction. The convergence order figure of – with and the one of – with h small enough are given in Figs. 1 and 2 under various mesh steps h and at . From Figs. 1 and 2, it is obvious that the scheme (11)–(14) is convergent in maximum norm, and the convergence order is Fig. 1


On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation.

Pan X, Zhang L - Springerplus (2016)

Temporal convergence order in maximal norm for  at  with different h and  computed by the scheme (11)–(14)
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4835426&req=5

Fig2: Temporal convergence order in maximal norm for at with different h and computed by the scheme (11)–(14)
Mentions: In computations, we always choose the parameter . Take the parameters . To verify the accuracy in the spatial direction, we take . And we choose h small enough to verify the second-order accuracy in the temporal direction. The convergence order figure of – with and the one of – with h small enough are given in Figs. 1 and 2 under various mesh steps h and at . From Figs. 1 and 2, it is obvious that the scheme (11)–(14) is convergent in maximum norm, and the convergence order is Fig. 1

Bottom Line: In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation.The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time.The unique solvability of numerical solutions is shown.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematics and Information Science, Weifang University, Weifang, 261061 China ; School of Management, Qufu Normal University, Rizhao, 276800 China.

ABSTRACT
In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation. The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical "energy" of the system and finite difference method. The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time. The unique solvability of numerical solutions is shown. A priori estimate and fourth-order convergence of the finite difference approximate solution are discussed by using discrete energy method and some techniques of matrix theory. Numerical results are given to show the validity and the accuracy of the proposed method.

No MeSH data available.