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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]

A partial semi-coarsening multigrid method based on the high-order compact (HOC) difference scheme on nonuniform grids is developed to solve the 2D convection–diffusion problems with boundary or internal layers. The significance of this study is that the multigrid method allows different number of grid points along different coordinate directions on nonuniform grids. Numerical experiments on some convection–diffusion problems with boundary or internal layers are conducted. They demonstrate that the partial semi-coarsening multigrid method combined with the HOC scheme on nonuniform grids, without losing the high-order accuracy, is very efficient and effective to decrease the computational cost by reducing the number of grid points along the direction which does not contain boundary or internal layers. [ABSTRACT FROM PUBLISHER]

An implicit–explicit Euler scheme in temporal direction is employed to discretize a partial integro-differential equation, which arises in pricing options under jump-diffusion process. Then the semi-discretized equation is approximated in space by the Sinc–Galerkin method with exponential accuracy. Meanwhile, the domain decomposition method is incorporated to handle the non-smoothness of the payoff function, and the improved fast Gauss transform is applied to accelerate the evaluation of the jump integral term. An effective preconditioner is proposed for solving the resulting dense Toeplitz-related systems by the preconditioned generalized minimum residual (GMRES) method. Numerical tests are performed to illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

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]