Fast and Robust Computation of the Matrix Absolute Value Function. Application to Roe Solver for the Numerical Simulation of Two-Phase Flow Models

The application of the generalized Roe scheme to the numerical simulation of two-phase flow models requires a fast and robust computation of the absolute value of the system matrix. In several models such as the two-fluid model or a general multi-field model, this matrix has a non trivial Eigen-structure and the Eigen decomposition is often ill conditioned. We give two general algorithms avoiding the diagonalization process: an iterative computation, which turns out to be an exact computation, and an interpolation algorithm which is faster and can handle the case of complex eigenvalues. The knowledge of the characteristic polynomial gives us an easy access to the eigenvalues but however, the iterative scheme can be used with only estimates of the eigenvalues, using for example Gershgorin's disk localization. We finally show some numerical results of two-fluid model simulations involving interfacial pressure and virtual mass force models. (authors)