525 research outputs found
High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problem
This is the peer reviewed version of the following article: Qiao L, Xu D, Yan Y. (2020). High-order ADI orthogonal spline collocation method for a new 2D fractional integro-differential problem. Mathematical Methods in the Applied Sciences, 1-17., which has been published in final form at https://doi.org/10.1002/mma.6258. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.We use the generalized L1 approximation for the Caputo fractional deriva-tive, the second-order fractional quadrature rule approximation for the inte-gral term, and a classical Crank-Nicolson alternating direction implicit (ADI)scheme for the time discretization of a new two-dimensional (2D) fractionalintegro-differential equation, in combination with a space discretization by anarbitrary-order orthogonal spline collocation (OSC) method. The stability of aCrank-Nicolson ADI OSC scheme is rigourously established, and error estimateis also derived. Finally, some numerical tests are give
Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations
We consider an initial-boundary value problem for
, that is, for a fractional
diffusion () or wave () equation. A numerical solution
is found by applying a piecewise-linear, discontinuous Galerkin method in time
combined with a piecewise-linear, conforming finite element method in space.
The time mesh is graded appropriately near , but the spatial mesh is
quasiuniform. Previously, we proved that the error, measured in the spatial
-norm, is of order , uniformly in , where
is the maximum time step, is the maximum diameter of the spatial finite
elements, and . Here,
we generalize a known result for the classical heat equation (i.e., the case
) by showing that at each time level the solution is
superconvergent with respect to : the error is of order
. Moreover, a simple postprocessing step
employing Lagrange interpolation yields a superconvergent approximation for any
. Numerical experiments indicate that our theoretical error bound is
pessimistic if . Ignoring logarithmic factors, we observe that the
error in the DG solution at , and after postprocessing at all , is of
order .Comment: 24 pages, 2 figure
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
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
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A fast two-level Strang splitting method for multi-dimensional spatial fractional Allen-Cahn equations with discrete maximum principle
In this paper, we study the numerical solutions of the multi-dimensional
spatial fractional Allen-Cahn equations. After semi-discretization for the
spatial fractional Riesz derivative, a system of nonlinear ordinary
differential equations with Toeplitz structure is obtained. For the sake of
reducing the computational complexity, a two-level Strang splitting method is
proposed, where the Toeplitz matrix in the system is split into the sum of a
circulant matrix and a skew-circulant matrix. Therefore, the proposed method
can be quickly implemented by the fast Fourier transform, substituting to
calculate the expensive Toeplitz matrix exponential. Theoretically, the
discrete maximum principle of our method is unconditionally preserved.
Moreover, the analysis of error in the infinite norm with second-order accuracy
is conducted in both time and space. Finally, numerical tests are given to
corroborate our theoretical conclusions and the efficiency of the proposed
method
Parallelization of implicit finite difference schemes in computational fluid dynamics
Implicit finite difference schemes are often the preferred numerical schemes in computational fluid dynamics, requiring less stringent stability bounds than the explicit schemes. Each iteration in an implicit scheme involves global data dependencies in the form of second and higher order recurrences. Efficient parallel implementations of such iterative methods are considerably more difficult and non-intuitive. The parallelization of the implicit schemes that are used for solving the Euler and the thin layer Navier-Stokes equations and that require inversions of large linear systems in the form of block tri-diagonal and/or block penta-diagonal matrices is discussed. Three-dimensional cases are emphasized and schemes that minimize the total execution time are presented. Partitioning and scheduling schemes for alleviating the effects of the global data dependencies are described. An analysis of the communication and the computation aspects of these methods is presented. The effect of the boundary conditions on the parallel schemes is also discussed
Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators
The positive definiteness of real quadratic forms with convolution structures
plays an important role in stability analysis for time-stepping schemes for
nonlocal operators.In this work, we present a novel analysis tool to handle
discrete convolution kernels resulting from variable-step approximations for
convolution operators. More precisely, for a class of discrete convolution
kernels relevant to variable-step time discretizations,we show that the
associated quadratic form is positive definite under some easy-to-check
algebraic conditions. Our proof is based on an elementary constructing strategy
using the properties of discrete orthogonal convolution kernels and
complementary convolution kernels. To the best of our knowledge, this is the
first general result on simple algebraic conditions for the positive
definiteness of variable-step discrete convolution kernels. Using the unified
theory, the stability for some simple non-uniform time-stepping schemes can be
obtained in a straightforward way.Comment: 21 pages,1 figur
An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver
We propose an efficient algorithm for the immersed boundary method on
distributed-memory architectures, with the computational complexity of a
completely explicit method and excellent parallel scaling. The algorithm
utilizes the pseudo-compressibility method recently proposed by Guermond and
Minev [Comptes Rendus Mathematique, 348:581-585, 2010] that uses a directional
splitting strategy to discretize the incompressible Navier-Stokes equations,
thereby reducing the linear systems to a series of one-dimensional tridiagonal
systems. We perform numerical simulations of several fluid-structure
interaction problems in two and three dimensions and study the accuracy and
convergence rates of the proposed algorithm. For these problems, we compare the
proposed algorithm against other second-order projection-based fluid solvers.
Lastly, the strong and weak scaling properties of the proposed algorithm are
investigated
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