scholarly journals Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

2021 ◽  
Vol 19 (1) ◽  
pp. 782-802
Author(s):  
Gang Dong ◽  
Zhichang Guo ◽  
Wenjuan Yao
Keyword(s):  
High Order Accuracy ◽  
Order Accuracy ◽  
Alternating Direction Implicit ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Time Stepping ◽  
Alternating Direction ◽  
Special Cases ◽  
Compact Adi Scheme ◽  
Convergence Of The Scheme

Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

scholarly journals Higher-Order Linear-Time Unconditionally Stable Alternating Direction Implicit Methods for Nonlinear Convection-Diffusion Partial Differential Equation Systems

2014 ◽  
Vol 136 (6) ◽  
Author(s):  
Oscar P. Bruno ◽  
Edwin Jimenez
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Time ◽  
High Order Accuracy ◽  
Implicit Methods ◽  
Partial Differential ◽  
Nonlinear Convection ◽  
Alternating Direction Implicit ◽  
Convection Diffusion ◽  
Alternating Direction

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.

scholarly journals Preservation of Fine Structures in PDE-Based Image Denoising

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Hakran Kim ◽  
Velinda R. Calvert ◽  
Seongjai Kim
Keyword(s):  
Numerical Methods ◽  
Image Denoising ◽  
Numerical Technique ◽  
Significant Loss ◽  
Implicit Time ◽  
Alternating Direction Implicit ◽  
Time Stepping ◽  
Alternating Direction ◽  
Fine Structures ◽  
Efficiency And Reliability

Image denoising processes often lead to significant loss of fine structures such as edges and textures. This paper studies various innovative mathematical and numerical methods applicable for conventional PDE-based denoising models. The method of diffusion modulation is considered to effectively minimize regions of undesired excessive dissipation. Then we introduce a novel numerical technique for residual-driven constraint parameterization, in order for the resulting algorithm to produce clear images whose corresponding residual is as free of image textures as possible. A linearized Crank-Nicolson alternating direction implicit time-stepping procedure is adopted to simulate the resulting model efficiently. Various examples are presented to show efficiency and reliability of the suggested methods in image denoising.

scholarly journals A new formulation of finite difference and finite volume methods for solving a space fractional convection–diffusion model with fewer error estimates

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reem Edwan ◽  
Shrideh Al-Omari ◽  
Mohammed Al-Smadi ◽  
Shaher Momani ◽  
Andreea Fulga
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Finite Volume ◽  
Difference Method ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
The Finite Difference Method ◽  
Volume Method

AbstractConvection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional order using the finite volume method and the finite difference method. In this context, we present an alternative way for estimating the space fractional derivative by utilizing the fractional Grünwald formula. The proposed methods are conditionally stable with second-order accuracy in space and first-order accuracy in time. Many comparisons are performed to display reliability and capability of the proposed methods. Furthermore, several results and conclusions are provided to indicate appropriateness of the finite volume method in solving the space fractional convection–diffusion equation compared with the finite difference method.

scholarly journals Entropy Schemes for One-Dimensional Convection-Diffusion Equations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Rongsan Chen
Keyword(s):  
Diffusion Equation ◽  
Hyperbolic Conservation Laws ◽  
Operator Splitting ◽  
Splitting Method ◽  
Order Accuracy ◽  
Central Difference ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Operator Splitting Method

In this paper, we extend the entropy scheme for hyperbolic conservation laws to one-dimensional convection-diffusion equation. The operator splitting method is used to solve the convection-diffusion equation that is divided into conservation and diffusion parts, in which the first-order accurate entropy scheme is applied to solve the conservation part and the second accurate central difference scheme is applied to solve the diffusion part. Numerical tests show that the L∞ error achieves about second-order accuracy, but the L1 error reaches about forth-order accuracy.

Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Convection-Diffusion Equation

2020 ◽  
Vol 20 (4) ◽  
pp. 709-715 ◽  
Author(s):  
Grigorii I. Shishkin ◽  
Lidia P. Shishkina
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Maximum Norm ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
First Order ◽  
Grid Approximation ◽  
Uniform Grids ◽  
Initial Boundary Value

AbstractThe convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x-derivative in the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval {(0,1]}. For small ε, the problem involves a boundary layer of width {\mathcal{O}(\varepsilon)}, where the solution changes by a finite value, while its derivative grows unboundedly as ε tends to zero. We construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives). Using a priori estimates, we show that such a scheme converges as {\{\varepsilon N\},N_{0}\to\infty} in the maximum norm with first-order accuracy in {\{\varepsilon N\}} and {N_{0}}; as {N,N_{0}\to\infty}, the convergence is conditional with respect toN, where {N+1} and {N_{0}+1} are the numbers of mesh points in x and t, respectively. We develop an improved difference scheme on uniform meshes using the grid approximation of the first x-derivative in the convective term by the central difference operator under the condition{h\leq m\varepsilon}, which ensures the monotonicity of the scheme; here m is some rather small positive constant. It is proved that this scheme converges in the maximum norm at a rate of {\mathcal{O}(\varepsilon^{-2}N^{-2}+N^{-1}_{0})}. We compare the convergence rate of the developed scheme with the known Samarskii scheme for a regular problem. It is found that the improved scheme (for {\varepsilon=1}), as well as the Samarskii scheme, converges in the maximum norm with second-order accuracy inx and first-order accuracy int.

scholarly journals Sinc-Galerkin method for solving the time fractional convection–diffusion equation with variable coefficients

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Juan Chen ◽  
MingZhu Li ◽  
Qiang Xu
Keyword(s):  
Diffusion Equation ◽  
Galerkin Method ◽  
Numerical Scheme ◽  
Variable Coefficients ◽  
Exponential Rate ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Exponential Rate Of Convergence ◽  
Time Fractional Derivative

Abstract In this paper, a new numerical algorithm for solving the time fractional convection–diffusion equation with variable coefficients is proposed. The time fractional derivative is estimated using the $L_{1}$ L 1 formula, and the spatial derivative is discretized by the sinc-Galerkin method. The convergence analysis of this method is investigated in detail. The numerical solution is $2-\alpha$ 2 − α order accuracy in time and exponential rate of convergence in space. Finally, some numerical examples are given to show the effectiveness of the numerical scheme.

scholarly journals A mass-conservative higher-order ADI method for solving unsteady convection–diffusion equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ben Wongsaijai ◽  
Nattakorn Sukantamala ◽  
Kanyuta Poochinapan
Keyword(s):  
Numerical Experiments ◽  
Mass Conservation ◽  
Cost Effective ◽  
Unconditionally Stable ◽  
Alternating Direction Implicit ◽  
Discrete Fourier Analysis ◽  
Convection Diffusion ◽  
Alternating Direction ◽  
Convection Dominated ◽  
And Diffusion

Abstract In the paper, a high-order alternating direction implicit (ADI) algorithm is presented to solve problems of unsteady convection and diffusion. The method is fourth- and second-order accurate in space and time, respectively. The resulting matrix at each ADI computation can be obtained by repeatedly solving a penta-diagonal system which produces a computationally cost-effective solver. We prove that the proposed scheme is mass-conserved and unconditionally stable by means of discrete Fourier analysis. Numerical experiments are performed to validate the mass conservation and illustrate that the proposed scheme is accurate and reliable for convection-dominated problems.

scholarly journals New High-Order Compact ADI Algorithms for 3D Nonlinear Time-Fractional Convection-Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuying Zhai ◽  
Xinlong Feng ◽  
Zhifeng Weng
Keyword(s):  
Diffusion Equation ◽  
Time Derivative ◽  
Three Dimensional ◽  
Cost Effective ◽  
Linearization Method ◽  
High Order ◽  
Thomas Algorithm ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Alternating Direction

Numerical approximations of the three-dimensional (3D) nonlinear time-fractional convection-diffusion equation is studied, which is firstly transformed to a time-fractional diffusion equation and then is solved by linearization method combined with alternating direction implicit (ADI) method. By using fourth-order Padé approximation for spatial derivatives and classical backward differentiation method for time derivative, two new high-order compact ADI algorithms with ordersO(τmin(1+α,2−α)+h4)andO(τ2−α+h4)are presented. The resulting schemes in each ADI solution step corresponding to a tridiagonal matrix equation can be solved by the Thomas algorithm which makes the computation cost effective. Numerical experiments are shown to demonstrate the high accuracy and robustness of two new schemes.

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