116 research outputs found
A Class of Second Order Difference Approximation for Solving Space Fractional Diffusion Equations
A class of second order approximations, called the weighted and shifted
Gr\"{u}nwald difference operators, are proposed for Riemann-Liouville
fractional derivatives, with their effective applications to numerically
solving space fractional diffusion equations in one and two dimensions. The
stability and convergence of our difference schemes for space fractional
diffusion equations with constant coefficients in one and two dimensions are
theoretically established. Several numerical examples are implemented to
testify the efficiency of the numerical schemes and confirm the convergence
order, and the numerical results for variable coefficients problem are also
presented.Comment: 24 Page
Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity
This study presents a novel high-order numerical method designed for solving
the two-dimensional time-fractional convection-diffusion (TFCD) equation. The
Caputo definition is employed to characterize the time-fractional derivative. A
weak singularity at the initial time () is encountered in the considered
problem, which is effectively managed by adopting a discretization approach for
the time-fractional derivative, where Alikhanov's high-order L2-1
formula is applied on a non-uniform fitted mesh, resulting in successful
tackling of the singularity. A high-order two-dimensional compact operator is
implemented to approximate the spatial variables. The alternating direction
implicit (ADI) approach is then employed to solve the resulting system of
equations by decomposing the two-dimensional problem into two separate
one-dimensional problems. The theoretical analysis, encompassing both stability
and convergence aspects, has been conducted comprehensively, and it has shown
that method is convergent with an order ,
where represents the order of the fractional derivative,
is the temporal discretization parameter and and represent spatial
mesh widths. Moreover, the parameter is utilized in the construction
of the fitted mesh.Comment: 21 page
HIGH-ORDER FINITE DIFFERENCE METHOD APPLIED TO THE SOLUTION OF THE THREE-DIMENSIONAL HEAT TRANSFER EQUATION AND TO THE STUDY OF HEAT EXCHANGERS
Numerical experiments for four test problems are carried out to demonstrate the performance of the present method and to compare it with the others classical methods. The numerical solutions obtained are compared with the analytical solution as well as the results by other numerical schemes with emphasis on the application involving heat exchange in a rectangular channel. It can be easily seen that the proposed method is simple to implement and very efficient
Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation
A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion
equation in the case where the Gr¨unwald-Letnikov definition is used for the time-fractional derivative. The stability
and convergence of the proposed Crank-Nicolson scheme are also analyzed. Finally, numerical examples are presented
to test that the numerical scheme is accurate and feasible
Higher-Order Linear-Time Unconditionally Stable Alternating Direction Implicit Methods for Nonlinear Convection-Diffusion Partial Differential Equation Systems
We introduce a class of alternating direction implicit (ADI) methods, based on approximate factorizations of backward differentiation formulas (BDFs) of order p≥2, for the numerical solution of two-dimensional, time-dependent, nonlinear, convection-diffusion partial differential equation (PDE) systems in Cartesian domains. The proposed algorithms, which do not require the solution of nonlinear systems, additionally produce solutions of spectral accuracy in space through the use of Chebyshev approximations. In particular, these methods give rise to minimal artificial dispersion and diffusion and they therefore enable use of relatively coarse discretizations to meet a prescribed error tolerance for a given problem. A variety of numerical results presented in this text demonstrate high-order accuracy and, for the particular cases of p=2,3, unconditional stability
Non-oscillatory Spatial Solutions Criterion for Convection-Diffusion Problem
The fact that the convection-diffusion problems are essential in nature is supported by the presence of such problems in vast number of applications in both science as well as engineering. Some of these applications involve the computational domain’s grid structure issues in the numerical experiment of fluid dynamics. The paper highlights the important role of convection-diffusion flow parameters in the construction of the grid structure. We propose the a priori criterion formulation to avoid non-oscillatory solutions which is based on both Peclet and grid numbers, and serves as a systematic approach in setting grid related parameters of interest. Aiming at a more efficient process in choosing grid structure for computational domain, the criterion functions as a standard which also eliminates heuristic process in the scalar concentration prediction. The test cases’ calculated results verify the consistency of the criterion
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