Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method

Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye

doi:10.1063/1.5023332

Cited by 10 publications

(12 citation statements)

References 15 publications

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“…First, we recall the original OEH method for the simple one dimensional equation form of Equation (1). The space and the time variable is discretized by setting nodes according to the most standard rules: x i = i∆x , i = 0, .…”

Section: The New Methodsmentioning

confidence: 99%

“…where r = αh ∆x 2 = − m ii h 2 > 0 , 0 < i < N − 1 is the usual mesh ratio and θ ∈ [0, 1]. For θ = 0, 1 2 , and 1 one obtains the implicit Euler, the Crank-Nicolson, and the explicit Euler (or, more concretely, the forward-time central-space, FTCS) schemes, respectively [39]. If θ < 1, the theta method is implicit.…”

Section: The New Methodsmentioning

confidence: 99%

“…We applied the following nine different values for parameter theta: θ ∈ {0, 1 5 , 1 4 , 1 3 , 1 2 , 2 3 , 3 4 , 4 5 , 1} in Equation (16). It means that, together with the CNe formula, we had 10 different formulas and we inserted all of these into the shifted-hopscotch structure in all possible combinations.…”

Section: Preliminary Testsmentioning

confidence: 99%

“…In the case of conductive heat transfer, u is the temperature and α = k/(cρ) is the thermal diffusivity, while k = k → r , t , c = c → r , t , and ρ = ρ → r , t refer to the following nonnegative quantities: heat conductivity, specific heat, and mass density, respectively. These equations and their generalizations, such as the diffusion-convectionreaction equation [1], can describe the diffusive transport of particles in very different physical, chemical, and biological systems, such as in semiconductor devices [2] and the human brain [3]. Furthermore, mathematically similar PDEs can be applied to simulate damped material flow through porous media, such as moisture [4], ground water, crude implicitness and we found no traces of trying to construct similar schemes while keeping the fully explicit property.…”

Section: Introductionmentioning

confidence: 99%

“…), S4 (0, 1 2 , 1 2 , 1 2 , 1), S5 (0, 1 2 , 1 2 , C, 1) In Figure 3, we present the error functions only for these top five combinations and some other, known methods, while in Figures 4 and 5 one can see the L ∞ and the energy errors vs. the total running times. Furthermore, Table 1 presents some results that were obtained by our numerical schemes, the two implementations of the Crank-Nicolson method and the "ode" routines of MATLAB.…”

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confidence: 99%

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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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Section: The New Methodsmentioning

confidence: 99%

Section: The New Methodsmentioning

confidence: 99%

Section: Preliminary Testsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Nagy

Saleh

Omle

et al. 2021

MCA

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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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