Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping

2022/10/28 by Baker, Katherine, Banjai, Lehel, Ptashnyk, Mariya · 4 citations
#35L77 #35R11 #65M06 #65M15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2210.16349

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 non-local 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 = 0 which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.

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