ABSTRACT
This chapter discuses the operator-splitting techniques that aimed at stabilizing the transport dominated systems without severe upstream-weighting error and at maintaining physical relevance by retaining the small physical diffusion/dispersion terms that give the correct physical entropy to the system of hyperbolic conservation laws. The objective of petroleum-related reservoir simulation, where supercomputers are playing a major role, is to understand the complex chemical, physical, and fluid flow and interactive processes occurring in a reservoir sufficiently well to be able to optimize the recovery of hydrocarbon. The equations are ill-conditioned and require effective preconditioners to accelerate convergence of iterative methods that are used predominantly in reservoir simulation. R. E. Ewing and R. F. Heinemann discussed finite-element weightings of the pressure obtained from standard finite-difference codes which resulted in significantly better phase velocities than via standard upstream-weighting methods.
