Efficient preconditioning of the method of lines for solving nonlinear twosided spacefractional diffusion equations
Moroney, Timothy J. & Yang, Qianqian (2012) Efficient preconditioning of the method of lines for solving nonlinear twosided spacefractional diffusion equations. In The 5th Symposium on Fractional Differentiation and its Applications, 1417 May, 2012, Hohai University, Nanjing, China.

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Abstract
A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the nonlocal nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear twosided spacefractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.
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ID Code:  50614 

Item Type:  Conference Paper 
Refereed:  Yes 
Keywords:  Nonlinear twosided spacefractional di�usion equation, method of lines, Jacobianfree NewtonKrylov, banded preconditioning 
Subjects:  Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) 
Divisions:  Current > Schools > School of Mathematical Sciences 
Deposited On:  27 May 2012 23:06 
Last Modified:  25 Mar 2013 13:21 
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