We consider an initial-boundary value problem for the n-dimensional wave equation with the variable sound speed, n\geq 1. We construct three-level implicit in time  compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for n=2) and three-parameter (for n=3) families. They are closely connected to some methods and schemesconstructed recently by several authors . In a unified manner, we prove the conditional stability of schemes in the strong and weak energy norms together with the 4th order error estimate under natural conditions on the time step. We also  give an example of extending a compact scheme for non-uniform in space and time rectangular meshes. We suggest simple effective iterative methods based on FFT to implement the schemes whose convergence rate, under the stability condition, is fast and independent on both the meshes and variable sound speed . A new effective initial guess to start iterations is given too. We also present promising results of numerical experiments.
We consider an initial-boundary value problem for the n-dimensional wave equation with the variable sound speed, n\geq 1. We construct three-level implicit in time and compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for n=2) and three-parameter (for n=3) families. We also show that some already known methods can be converted into such schemes . In a unified manner, we prove the conditional stability of schemes in the strong and weak energy norms together with the 4th order error estimate under natural conditions on the time step. We also transform an unconditionally stable 4th order two-level scheme suggested for n=2 to the three-level form, extend it for any n\geq 1 and prove its stability. We also give an example of a compact scheme for non-uniform in space and time rectangular meshes. We suggest simple fast iterative methods based on FFT to implement the schemes . A new effective initial guess to start iterations is given too. We also present promising results of numerical experiments.
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and
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They are closely connected to some methods and schemesconstructed recently by several authors
We also show that some already known methods can be converted into such schemes
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transform an unconditionally stable 4th order two-level scheme suggested for n=2 to the three-level form, extend it for any n\geq 1 and prove its stability. We also
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extending
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effective
fast
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whose convergence rate, under the stability condition, is fast and independent on both the meshes and variable sound speed