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The diffusive-viscous wave equation (DVWE) is widely used in seismicexploration since it can explain frequency-dependent seismic reflections in areservoir with hydrocarbons. Most of the existing numerical approximations forthe DVWE are based on domain truncation with ad hoc boundary conditions.However, this would generate artificial reflections as well as truncationerrors. To this end, we directly consider the DVWE in unbounded domains. Wefirst show the existence, uniqueness, and regularity of the solution of theDVWE. We then develop a Hermite spectral Galerkin scheme and derive thecorresponding error estimate showing that the Hermite spectral Galerkinapproximation delivers a spectral rate of convergence provided sufficientlysmooth solutions. Several numerical experiments with constant and discontinuouscoefficients are provided to verify the theoretical result and to demonstratethe effectiveness of the proposed method. In particular, We verify the errorestimate for both smooth and non-smooth source terms and initial conditions. Inview of the error estimate and the regularity result, we show the sharpness ofthe convergence rate in terms of the regularity of the source term. We alsoshow that the artificial reflection does not occur by using the present method.

Analysis and Hermite spectral approximation of diffusive-viscous wave equations in unbounded domains arising in geophysics