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On CSCS-based iteration method for tempered fractional diffusion equations. (English) Zbl 1373.65058
The authors use a CSCS (circulant and skew-circulant splitting) iteration method to solve linear systems arising from the discretization by finite difference methods for time fractional differential equations in one space dimension. The method is shown to be unconditionally convergent and the convergence rate is fast in numerical tests. In each iteration, a circulant system and a skew-circulant system are required to be solved which cost only \(O(N\log N)\) operations by the fast Fourier transform, where \(N\) is the number of interior mesh points in space. Moreover, the induced preconditioner possesses circulant-times-skew-circulant structure so that it can be inverted in \(O(N\log N)\) operation. Some numerical experiments are presented in which the preconditioner performs well with a simple choice of the method parameter.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65T50 Numerical methods for discrete and fast Fourier transforms
35K05 Heat equation
35R11 Fractional partial differential equations
65F08 Preconditioners for iterative methods
Software:
TFPDE
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