  We describe a method for removing the numerical errors in the modeling of
linear evolution equations that are caused by approximating the time derivative
by a finite difference operator. The method is based on integral transforms
realized as certain Fourier integral operators, called time dispersion
transforms, and we prove that, under an assumption about the frequency content,
it yields a solution with correct evolution throughout the entire lifespan. We
demonstrate the method on a model equation as well as on the simulation of
elastic and viscoelastic wave propagation.
