On properties of a semi-explicit vector compact scheme for the acoustic wave equation

A. A. Zlotnik, T. A. Lomonosov

Higher School of Economics University, Moscow, Russian Federation

Abstract: We numerically solved an initial-boundary value problem for the $n$-dimensional acoustic wave equation ($n\geqslant1$) with variable sound speed and nonhomogeneous Dirichlet boundary conditions. We studied a non-standard, three-level, semi-explicit compact scheme. The scheme uses three points per spatial direction and exploits $n$ auxiliary functions to approximate second-order non-mixed spatial derivatives. At the first time level, we applied a two-level scheme without using data derivatives. The scheme involves solving tridiagonal matrix systems in all $n$ spatial directions. We proved theorems on conditional stability and $4$th-order error bounds. Our $3\mathrm{D}$ experiments confirmed $4$th-order accuracy with minimal error, even on coarse meshes.

Keywords: acoustic wave equation, semi-explicit three-level vector scheme, compact scheme of the $4$-th order of accuracy, conditional stability, error bound.

DOI: 10.51790/2712-9942-2024-5-3-01