A flux-based modified method of characteristics.

*(English)*Zbl 0765.76057Summary: A flux-based modified method of characteristics (MMOC) methodology in 1D is described which has the following properties: unconditional stability (though explicit), exact answers for integer \(CFL\) (Courant) numbers, completely conservative (locally and globally) and able to utilize various flux limiters and various characteristic- (trajectory-) tracking algorithms. The use of characteristics based on cell-wise constant characteristic velocities results in considerable code simplification, and van Leer’s MUSCL [B. van Leer, J. Comput. Phys. 32, No. 1, 101- 136 (1979)] is an acurate and cost-effective flux limiter. For \(CFL>1\) the flux limiter is applied only to the non-integer part of \(CFL\), whereas the integer part is exact for constant velocities; therefore acuracy improves with larger \(CFL\). It is not a cheap algorithm, although explicit, because the operation count per time step increases with the integer part of \(CFL\), but it is much more acurate than the commonly used implicit upstream differencing. This flux-based MMOC method is well suited for groundwater flow calculations in which large local Courant numbers arise owing to grid clustering.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

65M25 | Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

conservation; unconditional stability; flux limiters; tracking algorithms; MUSCL; Courant numbers
PDF
BibTeX
XML
Cite

\textit{P. J. Roache}, Int. J. Numer. Methods Fluids 15, No. 11, 1259--1275 (1992; Zbl 0765.76057)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Computational Fluid Dynamics, Hermosa, Albuquerque, NM, 1976. |

[2] | Celia, Adv. Water Resources 13 pp 187– (1990) |

[3] | ’Solution of advection-dominated transport by Eulerian-Lagrangian methods using the backward methods of characteristics’, Ph.D. Thesis, Department of Civil Engineering, MIT, 1987. |

[4] | and , ’Large-scale simulation of miscible displacement by mixed and characteristic finite element methods’, in (ed.), Mathematical and Computational Methods in Seismic Exploration and Reservoir Modeling, SIAM, Philadelphia, PA, 1986, pp. 85-107. |

[5] | and , ’Efficient implementation of the modified method of characteristics in finite difference models of solute transport’, Proc. Conf. on Solving Ground Water Problems with Models, Indianapolis, IN. February 1989, National Water Well Association, pp. 483-491. |

[6] | Neuman, J. Comput. Phys. 41 pp 270– (1981) |

[7] | Neuman, Int. j. numer. methods eng. 20 pp 321– (1984) |

[8] | and , ’Eulerian-Lagrangian methods for advection-dispersion’, in , , , and (eds), Finite Elements in Water Resources, Vol. 4, Proc. Fourth Intern. Conf. in Hannover, Germany, Berlin, 1982, pp. 14.41-14.68. |

[9] | and , ’Finite element and finite difference methods for continuous flows in porous media’, in (ed.), Mathematics of Reservoir Simulation, SIAM, Philadelphia, PA, 1983, pp. 35-106. |

[10] | and , ’Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analyses’, J. Computational Mechanics, in review. · Zbl 0774.76058 |

[11] | Ewing, IMA J. Numerical Analysis |

[12] | and , ’Eulerian-Lagrangian analysis of pollutant transport in shallow water’, MIT, R. M. Parsons Laboratory, Technical Report 296, 1984. |

[13] | Lai, J. Hydraul. Eng., ASCE 114 pp 1074– (1988) |

[14] | Yang, Int. j. numer. methods fluids 12 pp 225– (1991) |

[15] | Holly, J. Hydraul. Div. ASCE 103 pp 1259– (1977) |

[16] | ’Galerkin time stepping along characteristics for Burgers’ equation’, in IMACS Transactions on Scientific Computation, Vol. 1, ed., North-Holland, Amsterdam, 1983, pp. 183-192. |

[17] | and , ’Simulation of miscible displacement using mixed methods and a modified method of characteristics’, Soc. Petrol. Eng. AIME, SPE 12241, 1983, p. 71. |

[18] | Dahle, Numer. Methods Partial Diff. Eqns. 6 pp 279– (1990) |

[19] | , and , ’Reservoir simulation using mixed methods, a modified method of characteristics, and local grid refinement’, Proc. Joint IMA/SPE European Conference on Mathematics of Oil Recovery, Cambridge Univ., 25-27 July, 1989, Oxford Univ. Press. |

[20] | Staniforth, Mon. Weather Rev. 119 pp 2206– (1991) |

[21] | ’Recent developments and problem areas in computational fluid dynamics’, in Lecture Notes in Mathematics Series, Vol. 461, Springer, Berlin, 1975, pp. 195-256. |

[22] | Courant, IBM J. 11 pp 215– (1967) · Zbl 0145.40402 |

[23] | Van Leer, Appl. Numer. Math. 2 pp 379– (1986) |

[24] | personal communication, 1991. |

[25] | Boris, J. Comput. Phys. 11 pp 38– (1973) |

[26] | Book, J. Comput. Phys. 18 pp 248– (1975) |

[27] | Zalesak, J. Comput. Phys. 31 pp 335– (1979) |

[28] | Zalesak, J. Comput. Phys. 40 pp 497– (1981) |

[29] | Roe, Ann. Rev. Fluid Mech. 18 pp 337– (1986) |

[30] | ’Some contributions to the modeling of discontinuous flows’, Lectures in Applied Mathematics, Vol. 22, American Math. Society, Providence, Rhode Island, 1985. |

[31] | Harten, J. Comput. Phys. 49 pp 357– (1983) |

[32] | Yee, J. Comput. Phys. 68 pp 151– (1987) |

[33] | Godunov, Mat.-sborn. 47 pp 357– (1959) |

[34] | Van Leer, J. Comput. Phys. 32 pp 101– (1979) |

[35] | ’Upwind and symmetric shock-capturing schemes’, NASA TM 89464, May 1987. |

[36] | ’A class of high-resolution explicit and implicit shock-capturing methods’, NASA TM-101088, February 1989. |

[37] | Harten, SIAM J. Numer. Anal. 24 pp 297– (1987) |

[38] | , and , ’Uniformly high order accurate essentially non-oscillatory schemes, 3’, NASA CR-178101, 1986. |

[39] | Colella, J. Comput. Phys. 54 pp 174– (1984) |

[40] | Leonard, Comput. Methods Appl. Mech. Eng. 88 pp 17– (1991) |

[41] | ’Computational fluid dynamics algorithms and codes developed for WIPP site simulations’, in and (eds), Computational Mechanics, Proc. Asian Pacific Conf. on Computational Mechanics, Hong Kong, December 1991, Vol. 2, Balkema, Rotterdam, 1991, pp. 1325-1336. |

[42] | ’Eulerian-Lagrangian localized adjoint methods for advection-dominated problems’, Proc. 13th Dundee Biennial Conf. on Numerical Analysis, Research Notes in Mathematical Series, Pitman, Vol. 228, pp. 206-228, (1990). |

[43] | McCormick, SIAM J. Numer. Anal. 27 pp 636– (1990) |

[44] | Harlow, Phys. Fluids 8 pp 2182– (1965) |

[45] | , and , ’The MAC method’, LASL Report No. LA-3425, Los Alamos Scientific Laboratory, Los Alamos, NM, 1966. |

[46] | and , ’SECO 2DH user’s manual, version 2.1’, SAND90-7096, Sandia National Laboratories, Albuquerque, NM, August 1991. |

[47] | and , ’Advanced CFD methodology for fast transients encountered in nonlinear combustion problems’, SBIR Phase I Final Report, CFD Research Corporation, Huntsville, AL, 1989. |

[48] | and , ’Flux-corrected transport in a moving grid’, J. Comput. Phys., accepted for publication. · Zbl 0796.76058 |

[49] | ’Intermediate boundary conditions for LOD, ADI, and approximate factorization methods’, ICASE Report No. 85-21, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, March 1985. |

[50] | ’Validation exercises of a one-dimensional flux-based modified method of characteristics’, in (ed.), Proc. IX Int. Conf. on Computational Methods in Water Resources,Denver, CO, June 1992, Computational Mechanics Publications, Southhampton, 1992. |

[51] | and , ’A characteristic-mixed method for contaminant transport and miscible displacement’, in (ed.), Proc. IX Int. Conf. on Computational Methods in Water Resources, Denver, CO, June 1992, Computational Mechanics Publications, Southhampton, 1992. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.