The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present method.

翻译：diffusive-viscos 浪形方程式(DVWE)在地震勘探中被广泛使用,因为它可以解释碳氢化合物储油库中依赖频率的地震反射。DVWE的现有数字近似值大多基于与特定边界条件的域间脱线。然而,这将产生人为反射和脱线错误。为此,我们直接将DVWE在无界域中考虑。我们首先显示DVWE解决方案的存在、独特性和规律性。我们随后开发了赫米特光谱Galerkin方案,并得出相应的误差估计值,显示赫米特光谱Galerkin近似光谱光谱光谱光谱光谱光谱光谱光谱光谱的光谱率提供了足够平稳的解决方案。提供了若干个不变和不连续的数值实验,以核实理论结果和显示拟议方法的有效性。特别是,我们核查了光滑和非移动源条件和初始条件的误差估计值。鉴于错误估计和规律性的结果,我们展示了当前源值的趋同率的精确率。我们还展示了不使用人工法的反射法。