A marching technique for nonseparable equations

Author:
Louis W. Ehrlich

Journal:
Math. Comp. **33** (1979), 881-890

MSC:
Primary 65F10; Secondary 65N20, 68C25

DOI:
https://doi.org/10.1090/S0025-5718-1979-0528045-9

MathSciNet review:
528045

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Abstract | References | Similar Articles | Additional Information

Abstract: A multiple-shooting marching technique is described which is applicable to arbitrary block tridiagonal matrices derived from nonseparable difference equations which are solved many times. Comparison with other methods on a particular problem shows the method to be competitive with respect to time and storage.

**[1]**O. AXELSSON,*On Preconditioning and Convergence Acceleration in Sparse Matrix Problems*, CERN 74-10, Geneva.**[2]**Randolph E. Bank and Donald J. Rose,*Marching algorithms for elliptic boundary value problems. I. The constant coefficient case*, SIAM J. Numer. Anal.**14**(1977), no. 5, 792–829. MR**0502000**, https://doi.org/10.1137/0714055

Randolph E. Bank,*Marching algorithms for elliptic boundary value problems. II. The variable coefficient case*, SIAM J. Numer. Anal.**14**(1977), no. 5, 950–970. MR**0502001**, https://doi.org/10.1137/0714064**[3]**Randolph E. Bank and Donald J. Rose,*Marching algorithms for elliptic boundary value problems. I. The constant coefficient case*, SIAM J. Numer. Anal.**14**(1977), no. 5, 792–829. MR**0502000**, https://doi.org/10.1137/0714055

Randolph E. Bank,*Marching algorithms for elliptic boundary value problems. II. The variable coefficient case*, SIAM J. Numer. Anal.**14**(1977), no. 5, 950–970. MR**0502001**, https://doi.org/10.1137/0714064**[4]**Paul Concus, Gene H. Golub, and Dianne P. O’Leary,*A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations*, Sparse matrix computations (Proc. Sympos., Argonne Nat. Lab., Lemont, Ill., 1975) Academic Press, New York, 1976, pp. 309–332. MR**0501821****[5]**L. W. EHRLICH,*Iterative vs. a Direct Method for Solving Fourth Order Elliptic Difference Equations*, Proc.-A. C. M. National Meeting, Los Angeles, Calif., 1966, pp. 29-35.**[6]**Louis W. Ehrlich,*The numerical solution of a Navier-Stokes problem in a stenosed tube: a danger in boundary approximations of implicit marching schemes*, Comput. & Fluids**7**(1979), no. 4, 247–256. MR**568753**, https://doi.org/10.1016/0045-7930(79)90009-4**[7]**A. C. HINDMARSH,*Solution of Block-Tridiagonal Systems of Linear Algebraic Equations*, Lawrence Livermore Laboratory, UCID-30150, 1977.**[8]**David S. Kershaw,*The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations*, J. Computational Phys.**26**(1978), no. 1, 43–65. MR**0488669****[9]**J. A. MANTEUFFEL,*An Iterative Method for Solving Nonsymmetric Linear Systems with Dynamic Estimation of Parameters*, Dept. of Computer Science, Univ. of Illinois at Urbana-Champaign, UIUCDCS-R-75-758, 1975.**[10]**J. A. Meijerink and H. A. van der Vorst,*An iterative solution method for linear systems of which the coefficient matrix is a symmetric 𝑀-matrix*, Math. Comp.**31**(1977), no. 137, 148–162. MR**0438681**, https://doi.org/10.1090/S0025-5718-1977-0438681-4**[10a]**I. J. PEARSON & B. KAPLAN,*Computer Time Comparison of Point and Block Successive Overtaxation*, Report AFIT-TR-70-6, Air Force Institute of Technology, School of Engineering, Wright-Patterson AFB, Ohio, 1970.**[11]**P. J. ROACHE,*A Direct Method for the Discretized Poisson Equation*, Sandia Report SC-RR-70-S79, 1971.**[12]**P. J. ROACHE, "Marching methods for elliptic problems: Part I",*Numerical Heat Transfer*, v. 1, 1978, pp. 1-25.**[13]**S. C. EISENSTAT, M. C. GURSKY, M. H. SCHULTZ & A. H. SHERMAN,*Yale Sparse Matrix Package*II.*The Nonsymmetric Codes*, Res. Report #114, Dept. of Comput. Sci., Yale Univ., 1977.**[14]**J. F. THOMPSON, F. C. THAMES & C. W. MASTIN, "Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary twodimensional bodies,"*J. Computational Phys.*, v. 15, 1974, pp. 299-319.**[15]**J. F. THOMPSON, F. C. THAMES & C. W. MASTIN,*Boundary-Fitted Curvilinear Coordinate Systems for Solution of Partial Differential Equations on Fields Containing any Number of Arbitrary Two-Dimensional Bodies*, NASA CR-2729, 1977.**[16]**D. M. YOUNG, JR. & L. J. HAYES,*Notes on the Conjugate Gradient Method*, CNA Report, Univ. of Texas, Austin.**[17]**David M. Young,*Iterative solution of large linear systems*, Academic Press, New York-London, 1971. MR**0305568**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0528045-9

Article copyright:
© Copyright 1979
American Mathematical Society