Fourth‐order compact difference schemes for 1D nonlinear Kuramoto–Tsuzuki equation. Issue 6 (10th April 2015)

• Record Type: Journal Article
• Authors: Hu, Xiuling; Chen, Shanzhen; Chang, Qianshun
• Is Part Of: Numerical methods for partial differential equations. Volume 31:Issue 6(2015)
• Page Start: 2080
• Page End: 2109
• Publication Date: 2015-04-10
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21979
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>In this article, first, we establish some compact finite difference schemes of fourth‐order for 1D nonlinear Kuramoto–Tsuzuki equation with Neumann boundary conditions in two boundary points. Then, we provide numerical analysis for one nonlinear compact scheme by transforming the nonlinear compact scheme into matrix form. And using some novel techniques on the specific matrix emerged in this kind of boundary conditions, we obtain the priori estimates and prove the convergence in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1qtxb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21979:num21979-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> norm. Next, we analyze the convergence and stability for one of the linearized compact schemes. To obtain the maximum estimate of the numerical solutions of the linearized compact scheme, we use the mathematical induction method. The treatment is that the convergence in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1qtzw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21979:num21979-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> norm is obtained as well as the maximum estimate, further the convergence in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1qv0z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21979:num21979-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> norm. Finally, numerical experiments demonstrate the theoretical results and show that one of the linearized compact schemes is more accurate, efficient and robust than the others and the previous. It is worthwhile that the compact difference methods presented here can be extended to 2D case. As an example, we present one nonlinear compact scheme for 2D Ginzburg–Landau equation and numerical tests show that the method is accurate and effective. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2080–2109, 2015</p> </abstract>