: A family of flux-continuous, locally conservative, finite-volume schemes has been developed for solving the general tensor pressure equation of petroleum reservoir-simulation on structured and unstructured grids. These schemes are control-volume distributed. The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full tensor flow approximation. The family of flux-continuous schemes is quantified by a quadrature parameterization. Improved convergence using the quadrature parameterization has been established for the family of flux-continuous schemes. When applying these schemes to strongly anisotropic heterogeneous media they can fail to satisfy a maximum principle (as with other FEM and finite-volume methods) and result in loss of solution monotonicity for high anisotropy ratios causing spurious oscillations in the numerical pressure solution. This paper investigates the use of flux-splitting techniques to solve the discrete system for the problems with high anisotropy ratios and improve monotonicity of the solution. Flux-splitting schemes are presented together with a series of numerical results for test-cases with strong anisotropy ratios. In all cases the resulting numerical pressure solutions are free of spurious oscillations.