A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

Abdulkadir, Y.A.: Comparison of finite difference schemes for the wave equation based on dispersion. J. Appl. Math. Phys. 3, 1544–1562 (2015)

Agut, C., Diaz, J., Ezziani, A.: High-order discretizations for the wave equation based on the modified equation technique. In: 10ème Congrès Français d’Acoustique, Lyon, France (2010)

Alford, R.M., Kelley, K.R., Boore, D.M.: Accuracy of finite-difference modeling of the acoustic wave equation. Geophysics 39(6), 834–842 (1974)

Babushka, I.M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34(6), 2392–2423 (1997)

Barucq, H., Calandra, H., Diaz, J., Ventimiglia, F.: High-order time discretization of the wave equation by Nabla-P scheme. ESAIM Proc. EDP Sci. 45, 67–74 (2014)

Bayliss, A., Goldstein, C.I., Turkel, E.: On accuracy conditions for the numerical computation of waves. J. Comput. Phys. 59, 396–404 (1985)

Britt, S., Tsynkov, S.V., Turkel, E.: A compact fourth order scheme for the Helmholtz equation in polar coordinates. J. Sci. Comput. 45, 26–47 (2010)

Britt, S., Tsynkov, S.V., Turkel, E.: Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes. Commun. Comput. Phys. 9, 520–541 (2011)

Britt, S., Tsynkov, S.V., Turkel, E.: A high order numerical method for the Helmholtz equation with non-standard boundary conditions. SIAM J. Sci. Comput. 35, A2255–A2292 (2013)

Britt, S., Tsynkov, S.V., Turkel, E.: Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials. J. Comput. Phys. 354, 26–42 (2018)

Chabassier, J., Imperiale, S.: Introduction and study of fourth order theta schemes for linear wave equations. J. Comput. Appl. Math. 245, 194–212 (2013)

Chabassier, J., Imperiale, S.: Fourth-order energy-preserving locally implicit time discretization for linear wave equations. Int. J. Numer. Methods Eng. 106, 593–622 (2016)

Ciment, M., Leventhal, S.H.: Higher order compact implicit schemes for the wave equation. Math. Comput. 29, 985–994 (1975)

Cohen, G.C.: Higher Order Numerical Methods for Transient Wave Equations. Springer, New York (2002)

Cohen, G.C., Joly, P.: Construction analysis of fourth-order finite difference schemes for the acoustic wave equation in nonhomogeneous media. SIAM J. Numer. Anal. 33(4), 1266–1302 (1996)

Dablain, M.A.: The application of high order differencing to the scalar wave equation. Geophysics 51(1), 54–66 (1986)

Das, S., Liaob, W., Guptac, A.: An efficient fourth-order low dispersive finite difference scheme for a 2-D acoustic wave equation. J. Comput. Appl. Math. 258(1), 151–167 (2014)

Deraemaeker, A., Babushka, I., Bouillard, P.: Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int. J. Numer. Methods Eng. 46(4), 471–499 (1999)

Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014)

Gilbert, J., Joly, P.: Higher order time stepping for second order hyperbolic problems and optimal CFL conditions. Partial Differ. Equ. 16, 67–93 (2008)

Gottlieb, D., Turkel, E.: Dissipative two-four methods for time dependent problems. Math. Comput. 30, 703–723 (1976)

Greenbaum, A.: Iterative Methods for Solving Linear systems. SIAM, Philadelphia (1997)

Gustafsson, B., Mossberg, E.: Time compact high order difference methods for wave propagation. SIAM J. Sci. Comput. 26, 259–271 (2004)

Hamilton, B., Bilbao, S.: Fourth order and optimized finite difference scheme for the 2-D wave equation. In: Proceedings of 16th International Conference on Digital Audio Effects (DAFx-13), Maynooth, Ireland, September 2–6 (2013)

Henshaw, W.: A high-order accurate parallel solver for Maxwell’s equations on overlapping grids. SIAM J. Sci. Comput. 28(5), 1730–1765 (2006)

Joly, P., Rogriguez, J.: Optimized higher order time discretization of second order hyperbolic problems: construction and numerical study. J. Comput. Appl. Math. 234(6), 1953–1961 (2010)

Kozdon, J.E., Wilcox, L.C.: Stable coupling of non-conforming high-order finite difference methods. SIAM J. Sci. Comput. 38(2), A923–A952 (2016)

Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24(3), 199–215 (1972)

Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, New York (1973)

Li, Z.: http://tinyurl.com/z5x7log or http://www.math.pku.edu.cn

Liang, H., Liu, M.Z., Lv, W.: Stability of theta-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments. Appl. Math. Lett. 23(2), 198–206 (2010)

Liao, W.Y.: On the dispersion, stability and accuracy of a compact higher-order finite difference scheme for 3D acoustic wave equation. J. Comput. Appl. Math. 270, 571–583 (2013)

Liao, W., Yong, P., Dastour, H., Huang, J.: Efficient and accurate numerical simulation of acoustic wave propagation in a 2D heterogeneous media. Appl. Math. Comput. 321, 385–400 (2018)

Mattsson, K., Ham, F., Iaccarino, G.: Stable boundary treatment for the wave equation on second-order form. J. Sci. Comput. 41(3), 366–383 (2009)

Medvinsky, M., Tsynkov, S., Turkel, E.: The method of difference potentials for the Helmholtz equation using compact high order schemes. J. Sci. Comput. 53(1), 150–193 (2012)

Nordström, J., Lundquist, T.: Summation-by-parts in time. J. Comput. Phys. 251, 487–499 (2013)

Shubin, G.R., Bell, J.B.: The stability of numerical boundary treatments for compact high-order finite-difference schemes. J. Comput. Phys. 108, 272–295 (1993)

Singer, I., Turkel, E.: High-order finite difference methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 163(1–4), 343–358 (1998)

Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)

Turkel, E., Gordon, D., Gordon, R., Tsynkov, S.: Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number. J. Comput. Phys. 232(1), 272–287 (2013)

Virta, K., Mattsson, K.: Acoustic wave propagation in complicated geometries and heterogenous media. J. Sci. Comput. 61, 90–118 (2014)

Wang, S., Kreiss, G.: Convergence of summation-by-parts finite difference methods for the wave equation. J. Sci. Comput. 71(1), 219–245 (2017)

Zeumi, A.: Fourth order symmetric finite difference schemes for the acoustic wave equation. BIT Numer. Math. 45(3), 627–651 (2005)