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.

翻译：针对非线性声学中重要的Westervelt方程，我们发展了一种数值方法，其中衰减由一类非局部时间算子表示。本文提出并分析了基于梯形法则与A-稳定卷积求积的时间半离散化格式。连续方程的存在性与正则性分析为半离散系统的稳定性与误差分析提供了依据。误差分析考虑了$t=0$处的奇异性问题，并通过数值方案中的修正项加以处理。广泛的数值实验验证了理论结果。