Journal of Scientific Computing (2019) 81:1181–1209 https://doi.org/10.1007/s10915-019-00970-x Compact High Order Accurate Schemes for the Three Dimensional Wave Equation F. Smith 1 · S. Tsynkov 1 · E. Turkel 2 Received: 21 February 2019 / Accepted: 1 May 2019 / Published online: 6 May 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract We construct a family of compact fourth order accurate finite difference schemes for the three dimensional scalar wave (d’Alembert) equation with constant or variable propagation speed. High order accuracy is of key importance for the numerical simulation of waves as it reduces the dispersion error (i.e., the pollution effect). The schemes that we propose are built on a stencil that has only three nodes in any coordinate direction or in time, which eliminates the need for auxiliary initial or boundary conditions. These schemes are implicit in time and conditionally stable. A particular scheme with the maximum Courant number can be chosen within the proposed class. The inversion at the upper time level is done by FFT for constant coefficients and multigrid for variable coefficients, which keeps the overall complexity of time marching comparable to that of a typical explicit scheme. Keywords Unsteady wave propagation · Fourth order accurate approximation · Small stencil · Cartesian grid · Implicit scheme · Multigrid methods Mathematics Subject Classification 65M06 · 65M12 · 65M22 We dedicate this paper to the memory of Professor Saul (Shalom) Abarbanel who provided mentorship and inspiration to a whole generation of students and colleagues. Work supported by the US Army Research Office (ARO) under Grant No. W911NF-16-1-0115 and the US–Israel Binational Science Foundation (BSF) under Grant No. 2014048. B S. Tsynkov tsynkov@math.ncsu.edu https://stsynkov.math.ncsu.edu F. Smith ffsmith@ncsu.edu E. Turkel turkel@tauex.tau.ac.il http://www.math.tau.ac.il/~turkel/ 1 Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA 2 School of Mathematical Sciences, Tel Aviv University, 69978 Ramat Aviv, Tel Aviv, Israel 123 Author's personal copy