1,422 research outputs found
An Alternating Direction Explicit Method for Time Evolution Equations with Applications to Fractional Differential Equations
We derive and analyze the alternating direction explicit (ADE) method for
time evolution equations with the time-dependent Dirichlet boundary condition
and with the zero Neumann boundary condition. The original ADE method is an
additive operator splitting (AOS) method, which has been developed for treating
a wide range of linear and nonlinear time evolution equations with the zero
Dirichlet boundary condition. For linear equations, it has been shown to
achieve the second order accuracy in time yet is unconditionally stable for an
arbitrary time step size. For the boundary conditions considered in this work,
we carefully construct the updating formula at grid points near the boundary of
the computational domain and show that these formulas maintain the desired
accuracy and the property of unconditional stability. We also construct
numerical methods based on the ADE scheme for two classes of fractional
differential equations. We will give numerical examples to demonstrate the
simplicity and the computational efficiency of the method.Comment: 25 pages, 1 figure, 7 table
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
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