Theoretical link in numerical shock thickness and shock-capturing dissipation

This paper presents a theoretical link between numerical shock thickness and shock-capturing dissipation. The link is derived rigorously from the compressible flow governing equations involving explicitly added shock-capturing numerical dissipation terms. The derivation employs only a natural assumption that the shock-capturing dissipation takes its maximum at the maximum velocity gradient location within the numerically diffused shock layer. Therefore, as long as this assumption is satisfied, the shock-capturing dissipation can take an arbitrary distribution in space. The derived theoretical link is verified through the numerical experiment of a 1D normal-shock problem, where the results agree well with the derived theory. Furthermore, by controlling the numerical shock thickness across the hierarchical Cartesian mesh boundary using the derived theoretical link, we demonstrate that the nonphysical errors associated with the mismatch in the shock thickness are significantly reduced.