Efficient approximation of the exponential operator for discrete 2D advection-diffusion problems

Numerical Linear Algebra with Applications

Volumn 10, Issue 3, 2003, Pages 271-289

  Bergamaschi, Luca ^a   Caliari, Marco ^a   Vianello, Marco ^a  

^a UNIVERSITY OF PADOVA   (Italy)

Author keywords

Advection diffusion problems; Chebyshev series; Exponential operator; Sparse matrices

Indexed keywords

EID: 0037253743     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.288     Document Type: Article

References (29)

1. Edwards WS, Tuckerman LS, Friesner RA, Sorensen DC. Krylov methods for the incompressible Navier-Stokes equations. Journal of Computational Physics 1994; 110:82-102.
2. Friesner RA, Tuckerman LS, Dornblaser BC, Russo TV. A method for exponential propagation of large system of stiff nonlinear differential equations. Journal of Scientific Computing 1989; 4:327-354.
3. Gallopoulos E, Saad Y. Efficient solution of parabolic equations by Krylov subspace methods. SIAM Journal on Scientific and Statistical Computing 1992; 13:1236-1264.
4. Hairer E, Bader G, Lubich C. On the stability of semi-implicit methods for ODEs, BIT. Numerical Mathematics 1982; 22:211-232.
5. Hochbruck M, Lubich JJ. On Krylov subspace approximations to the matrix exponential. SIAM Journal on Numerical Analysis 1997; 34:1911-1925.
6. Hockbruck M, Lubich C, Selhofer H. Exponential integrators for large system of differential equations. SIAM Journal on Scientific Computing 1998; 19:1552-1574.
7. Druskin VL, Knizhnerman LA. Two polynomial methods for calculating functions of symmetric matrices. Computational Mathematics and Mathematical Physics 1989; 6:112-121.
8. Knizherman LA. Computation of functions of unsymmetric matrices by means of Arnoldi's method. Computational Mathematics and Mathematical Physics 1991; 31:5-16.
9. Saad Y. Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM Journal on Numerical Analysis 1992; 29:209-228.
10. Sidje RB. EXPOKIT: Software package for computing matrix exponentials. ACM Association for Computing Machinery. Transactions on Mathematical Software 1998; 24:130-156.
11. Bergamaschi L, Vianello M. Efficient computation of the exponential operator for large, sparse, symmetric matrices. Numerical Linear Algebra with Applications 2000; 7:27-45.
12. Moret I, Novati P. The computation of functions of matrices by truncated Faber series. Numerical Functional Analysis and Optimization 2001; 22:697-719.
13. Novati P. Polynomial methods for the computation of functions of unsymmetric matrices. Ph.D. Thesis, Dip.to di Scienze Matematiche, Università di Trieste, Trieste, 2000.
14. Lehoucq RB, Sorensen DC, Yang C. ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicity Restarted Arnoldi Methods, 8 October 1997.
15. Manteuffel TA. The Tchebychev iteration for nonsymmetric linear systems. Numerische Mathematik 1977; 28:307-327.
16. Hockbruck M, Lubich C, Selhofer H. Adaptive procedure for estimation of parameter for the nonsymmetric Tchebychev iteration. Numerische Mathematik 1978; 28:187-208.
17. Eiermann M. On semiiterative methods generated by Faber polynomials. Numerische Mathematik 1989; 56:139-156.
18. Starke G, Varga RS. A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations. Numerische Mathematik 1993; 64:213-240.
19. Sleijpen GLG, van der Vorst HA. A Jacobi-Davidson method for linear eigenvalue problems. SIAM Journal on Matrix Analysis and Applications 1996; 17:401-425.
20. Curtiss JH. Faber polynomials and Faber series. The American Mathematical Monthly 1971; 78:577-596.
21. Markushevic AI. Theory of Functions of a Complex Variable. Prentice-Hall: Englewood Cliffs, NJ, 1967.
22. Smirnov VI, Lebedev NA. Functions of a Complex Variable, Constructive Theory. Iliffe Books: London, 1968.
23. Ellacott SW. Computation of Faber series with application to numerical polynomial approximation in the complex plane. Mathematics of Computation 1983; 40:575-587.
24. Caliari M. Computation of the exponential operator for large, sparse, nonsymmetric matrices. Laurea thesis, Università di Padova, 1999 (in Italian).
25. Stewart DE, Leyk TS. Error estimates for Krylov subspace approximations of matrix exponentials. Journal of Computational and Applied Mathematics 1996; 72:359-369.
26. Quarteroni A, Valli A. Numerical Approximation of Partial Differential Equations. Springer series in Computational Mathematics, vol. 23. Springer: Berlin, 1994.
27. Geddes KO. Near-minimax polynomial approximation in an elliptical region. SIAM Journal on Numerical Analysis 1978; 15:1225-1233.
28. Schaefer MJ. A polynomial based iterative method for linear parabolic equations. Journal of Computational and Applied Mathematics 1990; 29:35-50.
29. Sleijpen GLG, van der Vorst HA. An interpolatory approximation of the matrix exponential based on Faber polynomials. Journal of Computational and Applied Mathematics 2001; 131:361-380.