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For effectively solving the nonsymmetric and indefinite linear system originating from the discretized Helmholtz equation with complex wavenumber, we propose a two-parameter modified matrix splitting iteration method. We establish the asymptotic convergence theory for this method and demonstrate its convergence under certain conditions. The proposed iteration method leads to a new preconditioner that can be efficiently inverted by using the discrete sine transform. Numerical experiments are carried out to show that the new preconditioner significantly improves the convergence rate of the GMRES method, which results in effective preconditioned GMRES method for solving the Helmholtz equation of high wavenumber. [ABSTRACT FROM AUTHOR]

In this paper, we study the discretized linear systems arising from the space-fractional diffusion equations with piecewise continuous coefficients. Using the implicit finite difference scheme with the shifted Grünwald discretization, the resulting linear systems are Toeplitz-like which can be written as the sum of a scaled identity matrix and two diagonal-times-Toeplitz matrices. Standard circulant preconditioners and the existing approximate circulant-inverse preconditioner do not work for such Toeplitz-like linear systems since the discontinuous diffusion coefficients cannot be well approximated by interpolation polynomials. The main aim of this paper is to propose a new approximate circulant-inverse preconditioner to handle the fractional diffusion equations when the diffusion coefficients are piecewise continuous with finite jump discontinuities. Our idea is to approximate the eigenvalues of circulant matrices by the interpolation formula instead of approximating the diffusion coefficients as done by the existing algorithms. Therefore, the discontinuity of the diffusion coefficients does not influence the efficiency of the preconditioner. Theoretically, the spectra of the resulting preconditioned matrices are shown to be clustered around one, which can guarantee the fast convergence rate of the proposed preconditioner. Numerical examples are provided to demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper we develop a fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing. Instead of using the time-stepping methods, we develop the Laplace transform methods for solving the free-boundary fractional diffusion equations. By approximating the free boundary, the Laplace transform is taken on a fixed space region to replace discretizing the temporal variable. The hyperbola contour integral method is exploited to restore the option values. Meanwhile, the coefficient matrix has theoretically proven to be sectorial. Therefore, the highly accurate approximation by the fast Laplace transform method is guaranteed. The numerical results confirm that the proposed method outperforms the full finite difference methods in regard to the accuracy and complexity. [ABSTRACT FROM AUTHOR]

Under a jump-diffusion process, the option pricing function satisfies a partial integro-differential equation. A fourth-order compact scheme is used to discretize the spatial variable of this equation. The boundary value method is then utilized for temporal integration because of its unconditional stability and high-order accuracy. Two approaches, the local mesh refinement and the start-up procedure with refined step size, are raised to avoid the numerical malfunction brought by the nonsmooth payoff function. The GMRES method with a preconditioner which comes from the Crank-Nicolson formula is employed to solve the resulting large-scale linear system. Numerical experiments demonstrate the efficiency of the proposed method when pricing European and double barrier call options in the jump-diffusion model. [ABSTRACT FROM AUTHOR]