We are concerned with central differencing schemes for solving scalar hyperbolic conservation laws arising in the simulation of multiphase flows in heterogeneous porous media. We compare the Kurganov-Tadmor (KT) [3] semi-discrete central scheme with the Nessyahu-Tadmor (NT) [27] central scheme. The KT scheme uses more precise information about the local speeds of propagation together with integration over nonuniform control volumes, which contain the Riemann fans. These methods can accurately resolve sharp fronts in the fluid saturations without introducing spurious oscillations or excessive numerical diffusion. We first discuss the coupling of these methods with velocity fields approximated by mixed finite elements. Then, numerical simulations are presented for two-phase, two-dimensional flow problems in multi-scale heterogeneous petroleum reservoirs. We find the KT scheme to be considerably less diffusive, particularly in the presence of high permeability flow channels, which lead to strong restrictions on the time step selection; however, the KT scheme may produce incorrect boundary behavior.

In presence of linear diffusion and non-positive dispersion, we prove well-posedness of the nonlinear conservation equation u_t+f(u)_x=\eps u_xx -\del((u_xx)^2)_x. Then, as the right-hand perturbations vanish, we prove convergence of the previous solutions to the entropy weak solution of the hyperbolic conservation law u_t+f(u)_x=0.

Well-posedness of the initial value problem u_t+f(u)_x=\eps u_xx -\del((u_xx)^2)_x; u(x,0)=u_0(x). Then, as \eps, \del tend to 0, we prove convergence of the previous solutions to the entropy weak solution of the initial value problem u_t+f(u)_x=0; u(x,0)=u_0(x).

We consider a conservation law perturbed by a linear diffusion and a general form of non-positive dispersion. We prove the convergence of the corresponding solution to the entropy weak solution of the hyperbolic conservation law.