Anchored HLLD: A Riemann solver for a hypoelastic diffuse-interface model of compressible fluid-solid interaction

Abstract

We present an HLLD Riemann solver for a multicomponent hypoelastic diffuse-interface model of compressible fluid-solid interaction, with no separately computed interface and a deviatoric stress advanced by an objective hypoelastic rate law outside conservation-law form. The difficulty specific to this model is that the stress evolution is nonconservative and, unlike the familiar volume-fraction terms of Kapila-type models, cannot be left outside the Riemann solve once the shear waves are resolved. The shear coefficient $ρ(G+τ_{xx})$ sets the impedance of the resolved shear waves and so belongs to their eigenstructure. Anchored HLLD carries it inside the solve by evaluating the HLLD fan twice at each face, anchored in turn to each adjacent cell-centered state, with every cell advancing from the flux anchored to its own state. HLL, HLLC, and anchored HLLD form a three-tier hierarchy of increasing wave resolution in one framework. We prove local structural properties: consistency at the reconstruction order, exact species-mass conservation, and tangential-momentum and energy drift vanishing under refinement at the reconstruction's face-jump order on smooth fields, with one-dimensional tests illustrating these structural properties and demonstrating first-order drift convergence on discontinuous fields. Across one-dimensional tests and two-dimensional production simulations in the open-source solver MFC, anchored HLLD resolves both the material contact and the shear waves, HLLC the contact alone, and HLL neither sharply. Where refinement studies are shown, anchored HLLD converges toward exact or semi-analytic solutions. A three-dimensional shock-ellipsoid interaction demonstrates the solver with all six stress components active.