5 research outputs found
An implicit integration factor method for a kind of spatial fractional diffusion equations
A kind of spatial fractional diffusion equations in this paper are studied.
Firstly, an L1 formula is employed for the spatial discretization of the
equations. Then, a second order scheme is derived based on the resulting
semi-discrete ordinary differential system by using the implicit integration
factor method, which is a class of efficient semi-implicit temporal scheme.
Numerical results show that the proposed scheme is accurate even for the
discontinuous coefficients.Comment: 7 pages, 1 figure and 4 tables. This paper is accepted by the Second
International Conference on Physics, Mathematics and Statistics. It will be
published in Journal of Physics: Conference Serie
A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time-space fractional diffusion equation
A block lower triangular Toeplitz system arising from time-space fractional
diffusion equation is discussed. For efficient solutions of such the linear
system, the preconditioned biconjugate gradient stabilized method and flexible
general minimal residual method are exploited. The main contribution of this
paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is
developed for the block lower triangular Toeplitz system, whose storage is of
with being the spatial grid number; (ii) A new
skew-circulant preconditioner is designed to fast calculate the inverse of the
block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical
experiments are given to demonstrate the efficiency of our preconditioners.Comment: 19 pages, 3 figures, 5 tabl
A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation
An all-at-once linear system arising from the nonlinear tempered fractional
diffusion equation with variable coefficients is studied. Firstly, the
nonlinear and linearized implicit schemes are proposed to approximate such the
nonlinear equation with continuous/discontinuous coefficients. The stabilities
and convergences of the two schemes are proved under several suitable
assumptions, and numerical examples show that the convergence orders of these
two schemes are in both time and space. Secondly, a nonlinear all-at-once
system is derived based on the nonlinear implicit scheme, which may suitable
for parallel computations. Newton's method, whose initial value is obtained by
interpolating the solution of the linearized implicit scheme on the coarse
space, is chosen to solve such the nonlinear all-at-once system. To accelerate
the speed of solving the Jacobian equations appeared in Newton's method, a
robust preconditioner is developed and analyzed. Numerical examples are
reported to demonstrate the effectiveness of our proposed preconditioner.
Meanwhile, they also imply that such the initial guess for Newton's method is
more suitable.Comment: 10 tables, 2 figure
A second-order accurate implicit difference scheme for time fractional reaction-diffusion equation with variable coefficients and time drift term
An implicit finite difference scheme based on the - formula
is presented for a class of one-dimensional time fractional reaction-diffusion
equations with variable coefficients and time drift term. The unconditional
stability and convergence of this scheme are proved rigorously by the discrete
energy method, and the optimal convergence order in the -norm is
with time step and mesh size . Then, the
same measure is exploited to solve the two-dimensional case of this problem and
a rigorous theoretical analysis of the stability and convergence is carried
out. Several numerical simulations are provided to show the efficiency and
accuracy of our proposed schemes and in the last numerical experiment of this
work, three preconditioned iterative methods are employed for solving the
linear system of the two-dimensional case.Comment: 27 pages, 5 figures, 5 table
Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations
In this paper, fast numerical methods are established for solving a class of
time distributed-order and Riesz space fractional diffusion-wave equations. We
derive new difference schemes by the weighted and shifted
Grnwald formula in time and the fractional centered difference
formula in space. The unconditional stability and second-order convergence in
time, space and distributed-order of the difference schemes are analyzed. In
the one-dimensional case, the Gohberg-Semencul formula utilizing the
preconditioned Krylov subspace method is developed to solve the symmetric
positive definite Toeplitz linear systems derived from the proposed difference
scheme. In the two-dimensional case, we also design a global preconditioned
conjugate gradient method with a truncated preconditioner to solve the
discretized Sylvester matrix equations. We prove that the spectrums of the
preconditioned matrices in both cases are clustered around one, such that the
proposed numerical methods with preconditioners converge very quickly. Some
numerical experiments are carried out to demonstrate the effectiveness of the
proposed difference schemes and show that the performances of the proposed fast
solution algorithms are better than other numerical methods.Comment: 36 pages, 7 figures, 12 table