{"title":"Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping","authors":"Katherine Baker, Lehel Banjai, Mariya Ptashnyk","doi":"10.1090/mcom/3945","DOIUrl":null,"url":null,"abstract":"<p>We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of nonlocal in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">t = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3945","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of nonlocal in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at t=0t = 0 which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.
我们为非线性声学中的一个重要方程 Westervelt 方程开发了一种数值方法，该方程的衰减形式由一类非局部时间算子表示。基于梯形法则和 A 稳定卷积正交的时间半离散化方法得到了阐述和分析。连续方程的存在性和正则性分析为半离散系统的稳定性和误差分析提供了信息。误差分析包括考虑 t = 0 t = 0 处的奇异性，通过在数值方案中使用修正来解决这一问题。大量的数值实验证实了这一理论。
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