Comparative study on order-reduced methods for linear third-order ordinary differential equations

Abstract

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.

Keywords

third-order ordinary differential equation / order-reduced method / sinc discretization / preconditioner / Krylov subspace method

References

[1] Bai Z. Z. Structured preconditioners for nonsingular matrices of block two-by-two structures. Math Comp, 2006, 75: 791-815

[2] Bai Z. Z., Chan R. H., Ren Z. R. On sinc discretization and banded preconditioning for linear third-order ordinary differential equations. Numer Linear Algebra Appl, 2011, 18: 471-497

[3] Bai Z Z, Chan R H, Ren Z R. On order-reducible sinc discretization and blockdiagonal preconditioning methods for linear third-order ordinary differential equations. ICMSEC Tech Report, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 2012

[4] Bai Z. Z., Huang Y. M., Ng M. K. On preconditioned iterative methods for Burgers equations. SIAM J Sci Comput, 2007, 29: 415-439

[5] Bai Z. Z., Huang Y. M., Ng M. K. On preconditioned iterative methods for certain time-dependent partial differential equations. SIAM J Numer Anal, 2009, 47: 1019-1037

[6] Bai Z. Z., Ng M. K. Preconditioners for nonsymmetric block Toeplitz-like-plus-diagonal linear systems. Numer Math, 2003, 96: 197-220

[7] Ford W. F. A third-order differential equation. SIAM Rev, 1992, 34: 121-122

[8] Howes F. A. The asymptotic solution of a class of third-order boundary-value problems arising in the theory of thin film flows. SIAM J Appl Math, 1983, 43: 993-1004

[9] Lund J., Bowers K. Sinc Methods for Quadrature and Differential Equations, 1992, Philadelphia: SIAM

[10] Ng M. K. Fast iterative methods for symmetric sinc-Galerkin systems. IMA J Numer Anal, 1999, 19: 357-373

[11] Ng M. K., Bai Z. Z. A hybrid preconditioner of banded matrix approximation and alternating direction implicit iteration for symmetric sinc-Galerkin linear systems. Linear Algebra Appl, 2003, 366: 317-335

[12] Ng M. K., Potts D. Fast iterative methods for sinc systems. SIAM J Matrix Anal Appl, 2002, 24: 581-598

[13] Smith R. C., Bogar G. A., Bowers K. L., Lund J. The sinc-Galerkin method for fourthorder differential equations. SIAM J Numer Anal, 1991, 28: 760-788

[14] Stenger F. Numerical Methods Based on Sinc and Analytic Functions, 1993, New York: Springer-Verlag

[15] Tuck E. O., Schwartz L. W. A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev, 1990, 32: 453-469