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：我们为Westervelt方程式开发了一个数字方法,这是非线性声学中的一个重要方程式,其形式是减速由非本地时间操作器中的某一类非本地操作员表示。根据捕捉和稳定的演进二次曲线说明和分析了时间上的半分解。对连续方程式进行的存在和定期分析,为半分解系统的稳定性和误差分析提供了依据。错误分析包括考虑以美元=0美元为单位的单数,在数字方法中使用校正方法解决这个问题。广泛的数字实验证实了理论。