13,537 research outputs found
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
Numerical algorithm for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term
In this paper, an alternating direction implicit (ADI) difference scheme for
two-dimensional time-fractional wave equation of distributed-order with a
nonlinear source term is presented. The unique solvability of the difference
solution is discussed, and the unconditional stability and convergence order of
the numerical scheme are analysed. Finally, numerical experiments are carried
out to verify the effectiveness and accuracy of the algorithm.Comment: 25 pages, 3 figure
Finite difference schemes for multi-term time-fractional mixed diffusion-wave equations
The multi-term time-fractional mixed diffusion-wave equations (TFMDWEs) are
considered and the numerical method with its error analysis is presented in
this paper. First, a approximation is proved with first order accuracy to
the Caputo fractional derivative of order Then the
approximation is applied to solve a one-dimensional TFMDWE and an implicit,
compact difference scheme is constructed. Next, a rigorous error analysis of
the proposed scheme is carried out by employing the energy method, and it is
proved to be convergent with first order accuracy in time and fourth order in
space, respectively. In addition, some results for the distributed order and
two-dimensional extensions are also reported in this work. Subsequently, a
practical fast solver with linearithmic complexity is provided with partial
diagonalization technique. Finally, several numerical examples are given to
demonstrate the accuracy and efficiency of proposed schemes.Comment: approximation compact difference scheme distributed order fast
solver convergenc
Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation
Solutions of the Dirichlet and Robin boundary value problems for the
multi-term variable-distributed order diffusion equation are studied. A priori
estimates for the corresponding differential and difference problems are
obtained by using the method of the energy inequalities. The stability and
convergence of the difference schemes follow from these a priory estimates. The
credibility of the obtained results is verified by performing numerical
calculations for test problems.Comment: 21 pages, 6 tables. This version generalizes the previos on
Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law
Generalization of the heat conduction equation is obtained by considering the
system of equations consisting of the energy balance equation and
fractional-order constitutive heat conduction law, assumed in the form of the
distributed-order Cattaneo type. The Cauchy problem for system of energy
balance equation and constitutive heat conduction law is treated analytically
through Fourier and Laplace integral transform methods, as well as numerically
by the method of finite differences through Adams-Bashforth and
Gr\"{u}nwald-Letnikov schemes for approximation derivatives in temporal domain
and leap frog scheme for spatial derivatives. Numerical examples, showing time
evolution of temperature and heat flux spatial profiles, demonstrate
applicability and good agreement of both methods in cases of multi-term and
power-type distributed-order heat conduction laws
A conservation formulation and a numerical algorithm for the double-gyre nonlinear shallow-water model
We present a conservation formulation and a numerical algorithm for the
reduced-gravity shallow-water equations on a beta plane, subjected to a
constant wind forcing that leads to the formation of double-gyre circulation in
a closed ocean basin. The novelty of the paper is that we reformulate the
governing equations into a nonlinear hyperbolic conservation law plus source
terms. A second-order fractional-step algorithm is used to solve the
reformulated equations. In the first step of the fractional-step algorithm, we
solve the homogeneous hyperbolic shallow-water equations by the
wave-propagation finite volume method. The resulting intermediate solution is
then used as the initial condition for the initial-boundary value problem in
the second step. As a result, the proposed method is not sensitive to the
choice of viscosity and gives high-resolution results for coarse grids, as long
as the Rossby deformation radius is resolved. We discuss the boundary
conditions in each step, when no-slip boundary conditions are imposed to the
problem. We validate the algorithm by a periodic flow on an f-plane with exact
solutions. The order-of-accuracy for the proposed algorithm is tested
numerically. We illustrate a quasi-steady-state solution of the double-gyre
model via the height anomaly and the contour of stream function for the
formation of double-gyre circulation in a closed basin. Our calculations are
highly consistent with the results reported in the literature. Finally, we
present an application, in which the double-gyre model is coupled with the
advection equation for modeling transport of a pollutant in a closed ocean
basin
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
Petrov-Galerkin and Spectral Collocation Methods for distributed Order Differential Equations
Distributed order fractional operators offer a rigorous tool for mathematical
modelling of multi-physics phenomena, where the differential orders are
distributed over a range of values rather than being just a fixed
integer/fraction as it is in standard/fractional ODEs/PDEs. We develop two
spectrally-accurate schemes, namely a Petrov-Galerkin spectral method and a
spectral collocation method for distributed order fractional differential
equations. These schemes are developed based on the fractional Sturm-Liouville
eigen-problems (FSLPs). In the Petrov-Galerkin method, we employ fractional
(non-polynomial) basis functions, called \textit{Jacobi poly-fractonomials},
which are the eigenfunctions of the FSLP of first kind, while, we employ
another space of test functions as the span of poly-fractonomial eigenfunctions
of the FSLP of second kind. We define the underlying \textit{distributed
Sobolev space} and the associated norms, where we carry out the corresponding
discrete stability and error analyses of the proposed scheme. In the
collocation scheme, we employ fractional (non-polynomial) Lagrange interpolants
satisfying the Kronecker delta property at the collocation points.
Subsequently, we obtain the corresponding distributed differentiation matrices
to be employed in the discretization of the strong problem. We perform
systematic numerical tests to demonstrate the efficiency and conditioning of
each method
Numerical methods for nonlocal and fractional models
Partial differential equations (PDEs) are used, with huge success, to model
phenomena arising across all scientific and engineering disciplines. However,
across an equally wide swath, there exist situations in which PDE models fail
to adequately model observed phenomena or are not the best available model for
that purpose. On the other hand, in many situations, nonlocal models that
account for interaction occurring at a distance have been shown to more
faithfully and effectively model observed phenomena that involve possible
singularities and other anomalies. In this article, we consider a generic
nonlocal model, beginning with a short review of its definition, the properties
of its solution, its mathematical analysis, and specific concrete examples. We
then provide extensive discussions about numerical methods, including finite
element, finite difference, and spectral methods, for determining approximate
solutions of the nonlocal models considered. In that discussion, we pay
particular attention to a special class of nonlocal models that are the most
widely studied in the literature, namely those involving fractional
derivatives. The article ends with brief considerations of several modeling and
algorithmic extensions which serve to show the wide applicability of nonlocal
modeling.Comment: Revised/Improved version. 126 pages, 18 figures, review pape
Discontinuous Galerkin methods for fractional elliptic problems
We provide a mathematical framework for studying different versions of
discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville
fractional elliptic problems on a finite domain. The boundedness and stability
analysis of the primal bilinear form are provided. A priori error estimate
under energy norm and optimal error estimate under norm are obtained
for DG methods of the different formulations. Finally, the performed numerical
examples confirm the optimal convergence order of the different formulations
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