A maximum principle for multiphase flow models


  • K.A. Novikov Institute of Numerical Mathematics of RAS (INM RAS)




maximum principle, multi-phase flow, black oil model


Two maximum principles for several multi-phase flow models are formulated and proved. The first one is valid for phase saturations in an incompressible two-phase flow model with constant viscosities. The second one is valid for the global pressure in two- and three-phase flow models with constant viscosities and is also valid for phase pressures in the case of zero capillary pressure.

Author Biography

K.A. Novikov


  1. H. Weinberger, “Invariant Sets for Weakly Coupled Parabolic and Elliptic Systems,” Rend. Mat. 8, 295-310 (1975).
  2. X. Liu and X. Zhang, “The Weak Maximum Principle for a Class of Strongly Coupled Elliptic Differential Systems,” J. Funct. Anal. 263 (7), 1862-1886 (2012).
  3. G. Gripenberg, “On the Strong Maximum Principle for Degenerate Parabolic Equations,” J. Differ. Equ. 242 (1), 72-85 (2007).
  4. L. E. Payne and G. A. Philippin, “On Maximum Principles for a Class of Nonlinear Second-Order Elliptic Equations,” J. Differ. Equ. 1980. 37 (1). 39-48.
  5. J. I. Diaz and J. Hernández, “Global Bifurcation and Continua of Nonnegative Solutions for Some Nonlinear Elliptic Eigenvalue Type Problems,” in Contribuciones Matemáticas: Homenaje al Profesor Enrique Outerelo Domí nguez} (Complutense Univ., Madrid, 2004), pp. 161-170.
  6. Z. Chen, “Degenerate Two-Phase Incompressible Flow: I. Existence, Uniqueness and Regularity of a Weak Solution,” J. Differ. Equ. 171 (2), 203-232 (2001).
  7. Z. Chen, “Formulations and Numerical Methods of the Black Oil Model in Porous Media,” SIAM J. Numer. Anal. 38 (2), 489-514 (2000).
  8. H. Holden, N. H. Risebro, and A. Tveito, “Maximum Principles for a Class of Conservation Laws,” SIAM J. Appl. Math. 55 (3), 651-661 (1995).
  9. Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media (SIAM Press, Philadelphia, 2006).
  10. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Springer, New York, 1999).



How to Cite

Новиков К.А. A Maximum Principle for Multiphase Flow Models // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2017. 18. 138-145. doi 10.26089/NumMet.v18r211



Section 1. Numerical methods and applications