A large number of fluid mechanics problems are described in terms of a scalar equation representing overall mass balance, and a system of equations modeling linear momentum balance or transport of chemical constituents. Examples include the Navier-Stokes equations, the shallowwater equations, as well as miscible and immiscible flow in porous media. The overall mass balance equation must be solved for a properly defined global pressure, and we shall refer to it as the pressure equation. This equation is parabolic in general, but almost elliptic for nearly incompressible systems. The system of equations describing linear momentum balance or transport of the different chemical constituents is referred to as the transport equation. It is a hyperbolic system if diffusion-like effects are negligible. Depending on the physical problem at hand, diffusion-like terms may come from the effect of viscosity, capillarity, or physical diffusion. The pressure and transport equations are usually tightly coupled, and the solution to this system of equations naturally develops shocks and boundary layers. These sharp features entail the presence of high frequency components in the solution, which cannot be captured by the simulation grid. It is well known that, for many numerical methods, this lack of resolubility often leads to instability. We present a fairly general formulation for the numerical solution of this type of problems, which we understand in the general context of systems of nonlinear conservation laws. The method emanates from a multiscale decomposition into resolved (or grid) scales and unresolved (or subgrid) scales [1]. The multiscale split is invoked in a variational setting, which leads to a rigorous definition of a grid-scale problem and a subgrid-scale problem. A noteworthy feature of the formulation is that nonlinearity of the equations is retained at the time of invoking the multiscale split. The subscale problem is modeled by means of a residual-based algebraic approximation. The model requires the definition of a matrix of intrinsic time scales, which we design for different problems of interest. The method is applied to the simulation of three-phase flow in porous media, and the shallow-water equations. The proposed method yields stabilized, highly accurate solutions, and illustrates the potential of the formalism for the numerical simulation of these challenging problems.
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