Implicit time integration of mass conservation equations is necessary to maintain a practically feasible time step for problems with substantial variation in the local CFL number. The adaptive implicit method (AIM) is a mixed implicit-explicit scheme that lowers the computational cost by adaptively applying, in both space and time, implicit time integration to a small number of cells with large local CFL values. The resulting computational efficiency is utilized to improve the accuracy of the solution by employing high-order spatial discretization for both types of cells. We show that regardless of the order of accuracy, the AIM discretization is inconsistent, and that it introduces substantial numerical errors at the interfaces between explicit and implicit cells. We perform an error analysis of discretizations based on AIM to determine the error characteristics with respect to the order of accuracy, as well as the number and propagation rate of the implicit-explicit boundaries. We demonstrate that the linear AIM operators are monotone, nonoscillatory, and stable. We also show that the associated error normalized by $\Delta t$ is bounded for finite, stationary AIM boundaries but increases as a power law when the AIM boundaries move with the speed of the error pulse. We derive the conditions for which large errors and nonmonotonic solutions are obtained. A consistent version of AIM is proposed and compared with the standard AIM scheme. The error analysis and new AIM framework provide a basis for the design and implementation of AIM schemes for problems of practical interest.

MSC codes

  1. 15A15
  2. 15A09
  3. 15A23


  1. adaptive implicit method
  2. mixed discretization
  3. partial implicitness
  4. hybrid time integration
  5. numerical error analysis
  6. monotone discretization
  7. conservation laws
  8. porous media flow

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K. Aziz and A. Settari, Petroleum Reservoir Simulation, Elsevier, London, 1979.
G. A. Behie, P. A. Forsyth Jr., and P. H. Sammon, Adaptive implicit methods applied to thermal simulation, SPE Reservoir Eng., 2 (1987), pp. 596–598.
T. Belytschko and R. Mullen, Stability of explicit-implicit mesh partitions in time integration, Internat. J. Numer. Methods Engrg., 12 (1978), pp. 1575–1586.
T. Belytschko, H. J. Yen, and R. Mullen, Mixed methods for time integration, Comput. Methods Appl. Mech. Engrg., 17 (1978), pp. 259–275.
M. Blunt and B. Rubin, Implicit flux limiting schemes for petroleum reservoir simulation, J. Comput. Phys., 102 (1992), pp. 194–210.
K. H. Coats, L. K. Thomas, and R. G. Pierson, Compositional and black oil reservoir simulation, SPE Res. Eval. Eng., 1 (1998), pp. 372–379.
B. Cockburn, C. Johnson, C.-W. Shu, and E. Tadmor, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Math. 1697, Springer-Verlag, Berlin, 1998.
D. A. Collins, L. X. Nghiem, Y.-K. Li, and J. E. Grabonstotter, An efficient approach to adaptive-implicit compositional simulation with an equation of state, SPE Reservoir Eng., 7 (1992), pp. 259–264.
J. P. Collins, P. Colella, and H. M. Glaz, An implicit explicit Eulerian Godunov scheme for compressible flow, J. Comput. Phys., 116 (1995), pp. 195–211.
W. Dai and P. R. Woodward, A high-order iterative implicit-explicit hybrid scheme for magnetohydrodynamics, SIAM J. Sci. Comput., 19 (1998), pp. 1827–1846.
C. N. Dawson and T. F. Dupont, A finite difference domain decomposition algorithm for numerical solution of the heat equations, Math. Comp., 57 (1991), pp. 63–71.
C. N. Dawson and T. F. Dupont, Explicit/implicit, conservative domain decomposition procedures for parabolic problems based on block-centered finite differences, SIAM J. Numer. Anal., 31 (1994), pp. 1045–1061.
Q. Du, M. Mu, and Z. N. Wu, Efficient parallel algorithms for parabolic problems, SIAM J. Numer. Anal., 39 (2001), pp. 1469–1487.
K. Duraisamy and J. D. Baeder, Implicit scheme for hyperbolic conservation laws using nonoscillatory reconstruction in space and time, SIAM J. Sci. Comput., 29 (2007), pp. 2607–2620.
K. Duraisamy, J. D. Baeder, and J.-G. Liu, Concepts and application of time-limiters to high resolution schemes, J. Sci. Comput., 19 (2003), pp. 139–162.
J. A. Ekaterinaris, Performance of high-order-accurate, low-diffusion numerical schemes for compressible flow, AIAA J., 42 (2004), pp. 493–500.
P. A. Forsyth Jr., and P. H. Sammon, Practical considerations for adaptive implicit methods in reservoir simulation, J. Comput. Phys., 62 (1985), pp. 265–281.
E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer-Verlag, New York, 1996.
A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), pp. 357–393.
W. Hundsdorfer, Partially implicit BDF2 blends for convection dominated flows, SIAM J. Numer. Anal., 38 (2001), pp. 1763–1783.
W. Hundsdorfer and F. Jaffre, Implicit-explicit time stepping with spatial discontinuous finite elements, Appl. Numer. Math., 45 (2003), pp. 231–254.
D. D. Knight, A hybrid explicit-implicit numerical algorithm for the three-dimensional compressible Navier-Stokes equations, AIAA J., 22 (1984), pp. 1056–1063.
F. Kwok and H. A. Tchelepi, Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media, J. Comput. Phys., 227 (2007), pp. 706–727.
J. Liou and T. E. Tezduyar, Iterative adaptive implicit-explicit methods for flow problems, Internat. J. Numer. Methods Fluids, 11 (1990), pp. 867–880.
T. F. Russell, Stability analysis and switching criteria for adaptive implicit methods based on the CFL condition, in Proceedings of the Tenth SPE Symposium on Reservoir Simulation, Houston, TX, 1989, pp. 97–107.
J. S. Shang, Implicit-explicit method for solving the Navier-Stokes equations, AIAA J., 16 (1978), pp. 496–502.
H.-S. Shi and H.-L. Liao, Unconditional stability of corrected explicit-implicit domain decomposition algorithms for parallel approximation of heat equations, SIAM J. Numer. Anal., 44 (2006), pp. 1584–1611.
G. W. Thomas and D. H. Thurnau, Reservoir simulation using an adaptive implicit method, SPE J., 23 (1983), pp. 760–768.
K. T. Yoon and T. J. Chung, Three-dimensional mixed explicit-implicit generalized Galerkin spectral element methods for high-speed turbulent compressible flows, Comput. Methods Appl. Mech. Engrg., 135 (1996), pp. 343–367.
K. T. Yoon, S. Y. Moon, S. A. Garcia, G. W. Heard, and T. J. Chung, Flowfield-dependent mixed explicit-implicit (FDMEI) methods for high and low speed and compressible and incompressible flows, Comput. Methods Appl. Mech. Engrg., 151 (1998), pp. 75–104.
Y. Zhuang, An alternating explicit implicit domain decomposition method for the parallel solution of parabolic equations, J. Comput. Appl. Math., 206 (2007), pp. 549–566.
Y. Zhuang and X.-H. Sun, Stabilized explicit-implicit domain decomposition methods for the numerical solution of parabolic equations, SIAM J. Sci. Comput., 24 (2002), pp. 335–358.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 2890 - 2914
ISSN (online): 1095-7197


Submitted: 14 May 2008
Accepted: 14 April 2009
Published online: 3 July 2009

MSC codes

  1. 15A15
  2. 15A09
  3. 15A23


  1. adaptive implicit method
  2. mixed discretization
  3. partial implicitness
  4. hybrid time integration
  5. numerical error analysis
  6. monotone discretization
  7. conservation laws
  8. porous media flow



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