Multiscale-stabilized solutions to one-dimensional systems of conservation laws

We present a variational multiscale formulation for the numerical solution of one-dimensional systems of conservation laws. The key idea of the proposed formulation, originally presented by Hughes [Comput. Methods Appl. Mech. Engrg., 127 (1995) 387–401], is a multiple-scale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties, capable of accurately representing the sharp features of the solution. In the formulation developed herein, the multiscale split is invoked prior to any linearization of the equations. Special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity-capturing diffusion. The methodology is applied to the one-dimensional simulation of three-phase flow in porous media, and the shallow water equations. These numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear, nearly hyperbolic systems on very coarse grids. Application of the numerical formulation to multidimensional problems is presented in a forthcoming paper.