We construct a new compact semi-explicit three-level in time fourth-order finite-difference scheme for numerical solving the general multidimensional acoustic wave equation, where both the speed of sound and density of a medium are variable. The scheme is three-point in each spatial direction, has the truncation order $\mathcal{O}(|h|^4+h_t^4)$ and is easily implementable. It seems to be the first compact scheme with such properties for the equation under consideration. It generalizes a semi-explicit compact scheme developed and studied recently in the much simpler case of the variable speed of sound only. Numerical experiments confirm the high precision of the scheme and its fourth error order not only in the mesh $C$ norm but in the mesh $C^1$ norm as well.

翻译：本文构造了一种新的紧致半显式三层时间四阶有限差分格式，用于数值求解一般多维声波方程，其中介质的声速和密度均为变量。该格式在每个空间方向上为三点格式，具有截断误差阶 $\mathcal{O}(|h|^4+h_t^4)$ 且易于实现。这似乎是针对所考虑方程的第一个具有此类性质的紧致格式。该格式推广了近期在仅声速可变这一简单得多的情况下开发和研究的半显式紧致格式。数值实验证实了该格式的高精度及其四阶误差阶，不仅在网格 $C$ 范数下，在网格 $C^1$ 范数下亦是如此。