@Article{cmes.2013.096.409,
AUTHOR = {Cheng-Yu Ku},
TITLE = {A Novel Method for Solving Ill-conditioned Systems of Linear Equations with Extreme Physical Property Contrasts},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {96},
YEAR = {2013},
NUMBER = {6},
PAGES = {409--434},
URL = {http://www.techscience.com/CMES/v96n6/27043},
ISSN = {1526-1506},
ABSTRACT = {This paper proposes a novel method, named the dynamical Jacobianinverse free method (DJIFM), with the incorporation of a two-sided equilibrium algorithm for solving ill-conditioned systems of linear equations with extreme physical property contrasts. The DJIFM is based on the construction of a scalar homotopy function for transforming the vector function of linear or nonlinear algebraic equations into a time-dependent scalar function by introducing a fictitious time-like variable. The DJIFM demonstrated great numerical stability for solving linear or nonlinear algebraic equations, particularly for systems involving ill-conditioned Jacobian or poor initial values that cause convergence problems. With the incorporation of a newly developed two-sided equilibrium algorithm, the solution of layered problems with extreme contrasts in the physical property that are typically highly ill-conditioned can be solved. The proposed method was then adopted for the solution of several highly ill-conditioned numerical examples, including the linear Hilbert matrix, linear Vandermonde matrix, layered linear and nonlinear groundwater flow problems. The results revealed that using the DJIFM and the two-sided equilibrium algorithm can improve the convergence and increase the numerical stability for solving layered problems.},
DOI = {10.3970/cmes.2013.096.409}
}