A class of new stable, explicit methods to solve the non‐stationary heat equation

Kovács, Endre

doi:10.1002/num.22730

Cited by 25 publications

(33 citation statements)

References 38 publications

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“…However, approaches such as the finite difference (FDM) or finite element (FEM) methods require the full spatial discretization of the system; thus, they are computationally demanding. We explained in our previous papers [3,4,7], and also demonstrate in this paper, that the widely used conventional solvers, either explicit or implicit, have serious difficulties. The explicit methods are usually conditionally stable, so when the stiffness of the problem is high, very small time step sizes have to be used.…”

Section: Introductionmentioning

confidence: 59%

“…Suppose now that one needs to solve Equation (2) when the material properties are not constants but functions of the space variables and the mesh is possibly unstructured. We have shown in our previous papers [3,7] (based on, e.g., Chapter 5 of the book [18]) that Equation (3) can be generalized to arbitrary grids consisting of cells of various shapes and properties:…”

Section: The New Methodsmentioning

confidence: 99%

“…The other formula we use is derived from the constant-neighbor (CNe) method, which was introduced in our previous papers [7,19]. For the leapfrog-hopscotch method in the one-dimensional case, this can be written as follows:…”

Section: The New Methodsmentioning

confidence: 99%

“…= 'Tol' =10 3 , towards a small minimum value, for example, 'Tol' =10 −5 . In addition to these solvers, we also used the following methods for comparison purposes; the original UPFD, the CNe, the two-and three-stage linear-neighbor (LNe and LNe3) methods [7], and the Dufort-Frankel (DF) algorithm [17] (p. 120):…”

Section: Comparison With Other Methods For a Large Moderately Stiff S...mentioning

confidence: 99%

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Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

et al. 2021

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In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 105 different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.

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Section: Introductionmentioning

confidence: 59%

Section: The New Methodsmentioning

confidence: 99%

Section: The New Methodsmentioning

confidence: 99%

Section: Comparison With Other Methods For a Large Moderately Stiff S...mentioning

confidence: 99%

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Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

et al. 2021

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“…The other formula we used was the constant neighbor (CNe) method, which was introduced in our papers [42,43] and now briefly restated here. The starting point is Equation (3), where an approximation is made: When the new value of a variable u n+1 i was calculated, we neglected the fact that the neighbors u n i−1 and u n i+1 were also changing during the time step.…”

Section: The New Methodsmentioning

confidence: 99%

New Stable, Explicit, Shifted-Hopscotch Algorithms for the Heat Equation

Nagy

Saleh

Omle

et al. 2021

MCA

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Our goal was to find more effective numerical algorithms to solve the heat or diffusion equation. We created new five-stage algorithms by shifting the time of the odd cells in the well-known odd-even hopscotch algorithm by a half time step and applied different formulas in different stages. First, we tested 105 = 100,000 different algorithm combinations in case of small systems with random parameters, and then examined the competitiveness of the best algorithms by testing them in case of large systems against popular solvers. These tests helped us find the top five combinations, and showed that these new methods are, indeed, effective since quite accurate and reliable results were obtained in a very short time. After this, we verified these five methods by reproducing a recently found non-conventional analytical solution of the heat equation, then we demonstrated that the methods worked for nonlinear problems by solving Fisher’s equation. We analytically proved that the methods had second-order accuracy, and also showed that one of the five methods was positivity preserving and the others also had good stability properties.

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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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A class of new stable, explicit methods to solve the non‐stationary heat equation

Cited by 25 publications

References 38 publications

Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

New Stable, Explicit, Shifted-Hopscotch Algorithms for the Heat Equation

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

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