On a semi-explicit fourth-order vector compact scheme for the acoustic wave equation

Abstract We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove conditional stability and derive 4th-order error bound in the enlarged energy norm. It exploits additional sought functions which approximate 2nd-order non-mixed spatial derivatives of the exact solution. At the first time level, a similar two-level in time scheme is applied, without using derivatives of the data in the problem. No iterations are required to implement the scheme. We also present results of various 3D numerical experiments. They demonstrate a very high accuracy of the scheme, for smooth data, and its advantages in the error behavior over the classical explicit 2nd-order scheme, for nonsmooth data. An example of the wave in a layered medium (with a discontinuous speed of sound) initiated by the Ricker-type wavelet source function is also included.