A high-order discontinuous Galerkin (DG) method is presented for nonequilibrium multi-material ((Formula presented.)) flow with sharp interfaces. Material interfaces are reconstructed using the algebraic THINC approach, resulting in a sharp interface resolution. The system assumes stiff velocity relaxation and pressure nonequilibrium. The presented DG method uses Dubiner's orthogonal basis functions on tetrahedral elements. This results in a unique combination of sharp multimaterial interfaces and high-order accurate solutions in smooth single-material regions. A novel shock indicator based on the interface conservation condition is introduced to mark regions with discontinuities. Slope limiting techniques are applied only in these regions so that nonphysical oscillations are eliminated while maintaining high-order accuracy in smooth regions. A local projection is applied on the limited solution to ensure discrete closure law preservation. The effectiveness of this novel limiting strategy is demonstrated for complex three-dimensional multi-material problems, where robustness of the method is critical. The presented numerical problems demonstrate that more accurate and efficient multi-material solutions can be obtained by the DG method, as compared to second-order finite volume methods.

Obtaining stable and high-order numerical solutions for multi-material hydrodynamics is an open challenge. Although slope limiters are widely used to maintain monotonicity near discontinuities, typical limiting procedures violate closure laws at the discrete level when applied to multi-material hydrodynamics equations. Due to this, the high-order expansions of quantities related by the closure laws are no longer consistent. The commonly observed symptom of this consistency-violation is that the numerical method fails to maintain constant pressure and velocity across material interfaces. This leads to sub-optimal convergence rates for smooth multi-material problems as well. Specialized limiting procedures that satisfy consistency while maintaining conservation need to be developed for such equations. A novel procedure that re-instates consistency into slope-limited high-order discretizations applied to the multi-material hydrodynamics equations is presented here. Using simple examples, it is demonstrated that the presented method satisfies closure laws at the discrete level, while maintaining conservative properties of the high-order method. Furthermore, this procedure involves a projection step which relies on the compact basis of the underlying spatial discretization, i.e. for discontinuous schemes (viz. DG and FV) the projection is local, and does not involve global matrix solves. Comparisons with conventional approaches emphasizes the necessity of the consistent closure-law preserving limiting approach, in order to maintain design order of accuracy for smooth multi-material problems.