Conclusion

In principle, one shouldn't ignore the error caused by the time discretization of a finite-difference method; instead, use it to trade against the error from the spatial difference operator. Designing spatial finite-difference operators this way leads to spatially compact finite-difference stencils that achieve higher accuracy than schemes based on Taylor-series expansion (obviously). I think it's still an open question as to whether this method is better than a method that uses optimized spatial finite-difference coefficients that are not a function of the local velocity (Holberg's method). For spatially compact operators I think it's important to include the effect of the time discretization, and in this case the method presented here is the method of choice. For long spatial operators, which are used when one needs to use as few points per shortest wavelength as possible, the amount of storage required would probably be discouraging, but in principle, the method presented here is more accurate.