A Fourier pseudospectral method for the "good" Boussinesq equation with second‐order temporal accuracy. Issue 1 (24th June 2014)

• Record Type: Journal Article
• Authors: Cheng, Kelong; Feng, Wenqiang; Gottlieb, Sigal; Wang, Cheng
• Is Part Of: Numerical methods for partial differential equations. Volume 31:Issue 1(2015:Jan.)
• Page Start: 202
• Page End: 224
• Publication Date: 2014-06-24
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21899
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>In this article, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with a specially designed second‐order time‐stepping for the numerical solution of the "good" Boussinesq equation. Our analysis improves the existing results presented in earlier literature in two ways. First, a <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2cfrvq2f" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21899:num21899-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>ℓ</mml:mo><mml:mo>∞</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>, </mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>;</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> convergence for the solution and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2cfrvq30" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21899:num21899-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>ℓ</mml:mo><mml:mo>∞</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>, </mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>;</mml:mo><mml:msup><mml:mo>ℓ</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> convergence for the time‐derivative of the solution are obtained in this article, instead of the <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2cfrvpxq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21899:num21899-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>ℓ</mml:mo><mml:mo>∞</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>, </mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>;</mml:mo><mml:msup><mml:mo>ℓ</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> convergence for the solution and the <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2cfrvq0b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21899:num21899-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>ℓ</mml:mo><mml:mo>∞</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>, </mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>;</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> convergence for the time‐derivative, given in De Frutos, et al., Math Comput 57 (1991), 109–122. In addition, we prove that this method is unconditionally stable and convergent for the time step in terms of the spatial grid size, compared with a severe restriction time step restriction <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2cfrvprz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21899:num21899-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Δ</mml:mo><mml:mi>t</mml:mi><mml:mo>≤</mml:mo><mml:mi>C</mml:mi><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> required by the proof in De Frutos, et al., Math Comput 57 (1991), 109–122.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 202–224, 2015</p> </abstract>