A fitted operator finite difference method of lines for singularly perturbed parabolic convection–diffusion problems

Mbroh, Nana Adjoah; Munyakazi, Justin B.

doi:10.1016/j.matcom.2019.03.007

Cited by 27 publications

(10 citation statements)

References 21 publications

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“…Now consistency and convergence will be discussed considering the schemes as direct PDE solvers as usual [ 66 ] (Section 4.2). [ 67 ] Subscripts containing t or x denote differentiation with respect to the time or space variables, respectively. First we give the truncation errors of not only for the CLL, but for the CNe and LNe methods as well, since they are not given in the original paper.…”

Section: The Properties Of the Proposed Three‐stage Cll Methodsmentioning

confidence: 99%

A New Stable, Explicit, Third‐Order Method for Diffusion‐Type Problems

Kovács

Nagy

Saleh

2022

Advcd Theory and Sims

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This paper reports on a novel explicit numerical method for the spatially discretized diffusion or heat equation. After discretizing the space variables as in conventional finite difference methods, this method does not use a finite difference approximation for the time derivatives, it instead combines constant-neighbor and linear-neighbor approximations, which decouple the ordinary differential equations, thus they can be solved analytically. In the obtained three-stage method, the time step size appears in exponential form with negative coefficients in the final expression. This property guarantees unconditional stability, as it is shown using von Neumann stability analysis. It is also proved that the convergence of the method is third order in the time step size. After verification, by solving Fisher's and Huxley's equations, it is demonstrated that it works for nonlinear equations as well. The new algorithm is tested against widely used numerical solvers for cases where the media is strongly inhomogeneous. According to the results, the new method is significantly more effective than the traditional explicit or implicit methods, especially for extremely large stiff systems. It is believed that this new method is unique in the sense that it is the first unconditionally stable explicit method with third-order convergence.

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Section: The Properties Of the Proposed Three‐stage Cll Methodsmentioning

confidence: 99%

A New Stable, Explicit, Third‐Order Method for Diffusion‐Type Problems

Kovács

Nagy

Saleh

2022

Advcd Theory and Sims

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“…The local methods compute the solutions at particular points; in contrast, the global ones obtain the solutions overall the problem domain 6,7 . For example, the finite element and finite difference methods are local, 8‐13 while the spectral methods are global 14‐18 . The spectral methods gained importance due to their high convergence speed, accuracy, and applicability to either bounded or unbounded domains 16‐19 .…”

Section: Introductionmentioning

confidence: 99%

Accurate spectral algorithm for two‐dimensional variable‐order fractional percolation equations

Abdelkawy

Mahmoud

Abualnaja

et al. 2021

Math Methods in App Sciences

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A highly accurate spectral algorithm for (2 + 1) fractional percolation equations with variable order (VO‐FPEs) is considered. We propose a shifted Legendre–Gauss–Lobatto collocation (SL‐GL‐C) method in conjunction with shifted Chebyshev–Gauss–Radau collocation (SC‐GR‐C) method to solve the two‐dimensional VO‐FPEs. A complete theoretical formulation is presented, and numerical results are given to illustrate the performance and efficiency of the algorithm. The superiority of the scheme to tackle VO‐FPEs is revealed, even when dealing with non‐smooth time solutions.

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“…The local methods listed the approximate solution at specific points, while the global methods give the approximate solution in whole the mentioned interval. The numerical approximations for differential equations [1][2][3][4] are listed at specific points using finite difference methods. While the finite element methods subdivide the whole interval into subintervals and give the approximate solution in them.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Technique for Riesz Fractional Chen-Lee-Liu Equation

Abdelkawy

Alyami

2021

Journal of Function Spaces

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This paper discusses the study of optical solitons that are modeled by Riesz fractional Chen-Lee-Liu model, one of the versions of the famous nonlinear Schrödinger equation. This model is solved by the assistance of consecutive spectral collocation technique with two independent approaches. The first is the approach of the spatial variable, while the other is the approach of the temporal variable. It is concluded that the method of the current paper is far more efficient and credible for the proposed problem. Numerical results illustrate the performance efficiency of the algorithm. The results also point out that the scheme can lead to spectral accuracy of the studied model.

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A fitted operator finite difference method of lines for singularly perturbed parabolic convection–diffusion problems

Cited by 27 publications

References 21 publications

A New Stable, Explicit, Third‐Order Method for Diffusion‐Type Problems

A New Stable, Explicit, Third‐Order Method for Diffusion‐Type Problems

Accurate spectral algorithm for two‐dimensional variable‐order fractional percolation equations

A Spectral Collocation Technique for Riesz Fractional Chen-Lee-Liu Equation

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