Efficient Solution of Exact Riemann Problems for Compressible Multiphase Flow and Fluid-Structure Interaction Simulations

When solving compressible multiphase flow problems, it is often important to account for the discontinuity of the equations of state (EOS) across material interfaces. One method to achieve this, known as FIVER ("FInite Volume method based on Exact multiphase Riemann solvers"), is to construct and solve an exact one-dimensional bimaterial Riemann problem between each pair of neighboring cells separated by a material interface. In this talk, we present a method to accelerate the solution of bimaterial Riemann problems with arbitrary (convex) EOS, based on the idea of storing and reusing previous solutions. The talk will start with a summary of the FIVER framework, including the use of level set method for tracking fluid-fluid interface. Then, the iterative solution of bimaterial Riemann problems will be discussed in detail, focusing on computation efficiency. For an arbitrary EOS, the Riemann invariants associated with rarefaction waves involve ordinary differential equations (ODEs) that cannot be solved analytically. The numerical solution of these ODEs significantly increases the computational cost. To mitigate this issue, we propose an acceleration technique that utilizes the results of previously solved Riemann problems. Specifically, in each simulation, an R-tree is constructed to store the inputs and outputs of bimaterial Riemann problems. When solving a new Riemann problem, a nearest neighbor search is performed using the R-tree to find data points that are closest to the current one. The corresponding outputs are then interpolated to provide an accurate initial guess for the current Riemann problem, thereby reducing the number of iterations required to achieve convergence. Finally, several numerical tests will be presented to demonstrate the acceleration effects of the proposed method in solving challenging multi-physics problems in the contexts of underwater explosion, laser lithotripsy, and hypervelocity impact.