Numerical solution of the equations governing multiphase porous media flow is challenging. A common approach to improve the performance of iterative non-linear solvers for these problems is to introduce artificial diffusion. Here, we present a mass conservative artificial diffusion that accelerates the non-linear solver but vanishes when the solution is converged. The vanishing artificial diffusion term is saturation dependent and is larger in regions of the solution domain where there are steep saturation gradients. The non-linear solver converges more slowly in these regions because of the highly non-linear nature of the solution. The new method provides accurate results while significantly reducing the number of iterations required by the non-linear solver. It is particularly valuable in reducing the computational cost of highly challenging numerical simulations, such as those where physical capillary pressure effects are dominant. Moreover, the method allows converged solutions to be obtained for Courant numbers that are at least two orders of magnitude larger than would otherwise be possible.