Optimized composite finite difference schemes for atmospheric flow modeling

Appadu, Appanah Rao

doi:10.1002/num.22407

Cited by 4 publications

(3 citation statements)

References 32 publications

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“…Our methods also bear some resemblance to the composite methods introduced and used by Fromm [43] and others [44–47] for hyperbolic equations such as the advection equation to reduce dissipative and dispersive errors at the same time. They usually apply different known methods such as the Lax–Wendroff and leapfrog in subsequent time steps, or the linear combination of these schemes, thus these methods are also different from ours.…”

Section: The Properties Of the New Three‐stage Methodsmentioning

confidence: 88%

A new stable, explicit, and generic third‐order method for simulating conductive heat transfer

Kovács

Nagy

2022

Numerical Methods Partial

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In this paper we introduce a new type of explicit numerical algorithm to solve the spatially discretized linear heat or diffusion equation. After discretizing the space variables as in standard finite difference methods, this novel method does not approximate the time derivatives by finite differences, but use three stage constant-neighbor and linear neighbor approximations to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee unconditional stability. The scheme contains a free parameter p. We show that the convergence of the method is third-order in the time step size regardless of the values of p, and, according to von Neumann stability analysis, the method is stable for a wide range of p. We validate the new method by testing the results in a case where the analytical solution exists, then we demonstrate the competitiveness by comparing its performance with several other numerical solvers.

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Section: The Properties Of the New Three‐stage Methodsmentioning

confidence: 88%

A new stable, explicit, and generic third‐order method for simulating conductive heat transfer

Kovács

Nagy

2022

Numerical Methods Partial

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“…However, in the case of those methods matrix exponentials are calculated, while we do not even need to use matrices at all, thus there are crucial differences compared to our methods. The new methods also bear some resemblance to the composite methods introduced by Fromm [ 53 ] more than five decades ago, which were also used by others [ 54–57 ] for hyperbolic equations, such as the advection equation, to reduce dissipative and dispersive errors at the same time. Composite in their case usually means that different known methods such as the Lax–Wendroff and leapfrog or the linear combination of these are applied in subsequent time steps, thus these methods are also different from ours.…”

Section: Introductionmentioning

confidence: 89%

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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“…For its part, other advances have been made in designing and applying finite difference schemes, including the corresponding stability analysis of these schemes. Appadu presents in [6,7] different schemes to numerically solve the 1D advection-diffusion equation with very satisfactory results. In addition, the work presented in [8] shows a complete stability analysis, obtaining stability regions of each of the proposed methods.…”

Section: Introductionmentioning

confidence: 99%

Study of the stability of a meshless generalized finite difference scheme applied to the wave equation

Tinoco-Guerrero

Domínguez-Mota

Guzmán-Torres

et al. 2023

Front. Appl. Math. Stat.

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When designing and implementing numerical schemes, it is imperative to consider the stability of the applied methods. Prior research has presented different results for the stability of generalized finite-difference methods applied to advection and diffusion equations. In recent years, research has explored a generalized finite-difference approach to the advection-diffusion equation solved on non-rectangular and highly irregular regions using convex, logically rectangular grids. This paper presents a study on the stability of generalized finite difference schemes applied to the numerical solution of the wave equation, solved on clouds of points for highly irregular domains. The stability analysis presented in this work provides significant insights into the proper discretizations needed to obtain stable and satisfactory results. The proposed explicit scheme is conditionally stable, while the implicit scheme is unconditionally stable. Notably, the stability analyses presented in this paper apply to any scheme which is at least second order in space, not just the proposed approach. The proposed scheme offers effective means of numerically solving the wave equation, particularly for highly irregular domains. By demonstrating the stability of the scheme, this study provides a foundation for further research in this area.

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Optimized composite finite difference schemes for atmospheric flow modeling

Cited by 4 publications

References 32 publications

A new stable, explicit, and generic third‐order method for simulating conductive heat transfer

A new stable, explicit, and generic third‐order method for simulating conductive heat transfer

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

Study of the stability of a meshless generalized finite difference scheme applied to the wave equation

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