In this paper, we introduce a hyperbolic model for entropy dissipative system of viscous conservation laws via a flux relaxation approach. We develop numerical schemes for the resulting hyperbolic relaxation system by employing the finite-volume methodology used in the community of hyperbolic conservation laws, e.g., the generalized Riemann problem method. For fully discrete schemes for the relaxation system of scalar viscous conservation laws, we show the asymptotic preserving property in the coarse regime without resolving the relaxation scale and prove the dissipation property by using the modified equation approach. Further, we extend the idea to the compressible Navier-Stokes equations. Finally, we display the performance of our relaxation schemes by a number of numerical experiments.

翻译：本文通过通量松弛方法，为粘性守恒律的熵耗散系统引入了一个双曲模型。我们利用双曲守恒律领域常用的有限体积方法（例如广义黎曼问题方法）为所得双曲松弛系统开发了数值格式。针对标量粘性守恒律松弛系统的全离散格式，我们证明了在不解析松弛尺度的粗网格区域内的渐近保持性质，并通过修正方程方法验证了耗散性质。进一步地，我们将该思想推广至可压缩Navier-Stokes方程。最后，通过一系列数值实验展示了所提松弛格式的性能。