Estimates of the hyperbolization effect on the heat equation

Myshetskaya, E. E.; Тишкин, В. Ф.

doi:10.1134/s0965542515080138

Cited by 20 publications

(5 citation statements)

References 8 publications

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“…It is legitimate to ask the question how far the solutions u h (x, t) of the hyperbolic equation (3.7) are from the solutions u p (x, t) of the parabolic diffusion equation (3.1) (for the same initial condition). This question for the initial value problem was studied in [20] and we shall provide here only the obtained error estimate. Let us introduce the difference between two solutions: δu (x, t)…”

Section: Error Estimatementioning

confidence: 99%

Stable explicit schemes for simulation of nonlinear moisture transfer in porous materials

Gasparin

Berger

Dutykh

et al. 2017

Journal of Building Performance Simulation

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Implicit schemes have been extensively used in building physics to compute the solution of moisture diffusion problems in porous materials for improving stability conditions. Nevertheless, these schemes require important sub-iterations when treating nonlinear problems. To overcome this disadvantage, this paper explores the use of improved explicit schemes, such as Dufort-Frankel, Crank-Nicolson and hyperbolisation approaches. A first case study has been considered with the hypothesis of linear transfer. The Dufort-Frankel, Crank-Nicolson and hyperbolisation schemes were compared to the classical Euler explicit scheme and to a reference solution. Results have shown that the hyperbolisation scheme has a stability condition higher than the standard Courant-Friedrichs-Lewy (CFL) condition. The error of this schemes depends on the parameter τ representing the hyperbolicity magnitude added into the equation. The Dufort-Frankel scheme has the advantages of being unconditionally stable and is preferable for nonlinear transfer, which is the three others cases studies. Results have shown the error is proportional to O(∆t) . A modified Crank-Nicolson scheme has been also studied in order to avoid sub-iterations to treat the nonlinearities at each time step. The main advantages of the Dufort-Frankel scheme are (i) to be twice faster than the Crank-Nicolson approach; (ii) to compute explicitly the solution at each time step; (iii) to be unconditionally stable and (iv) easier to parallelise on high-performance computer systems. Although the approach is unconditionally stable, the choice of the time discretisation ∆t remains an important issue to accurately represent the physical phenomena.

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Section: Error Estimatementioning

confidence: 99%

Stable explicit schemes for simulation of nonlinear moisture transfer in porous materials

Gasparin

Berger

Dutykh

et al. 2017

Journal of Building Performance Simulation

View full text Add to dashboard Cite

show abstract

“…It is legitimate to ask the question how far are solutions u h (x, t) to the hyperbolic equation (3.6) from the solutions u p (x, t) of the parabolic heat equation (1.1) (for the same initial condition). This question for the initial value problem was studied in [7] and we shall provide here only the obtained error estimate. Let us introduce the difference between two solutions: δu (x, t)…”

Section: Error Estimatementioning

confidence: 99%

How to overcome the Courant-Friedrichs-Lewy condition of explicit discretizations?

Dutykh

2016

Preprint

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This manuscript contains some thoughts on the discretization of the classical heat equation. Namely, we discuss the advantages and disadvantages of explicit and implicit schemes. Then, we show how to overcome some disadvantages while preserving some advantages. However, since there is no free lunch, there is a price to pay for any improvement in the numerical scheme. This price will be thoroughly discussed below.In particular, we like explicit discretizations for the ease of their implementation even for nonlinear problems. Unfortunately, when these schemes are applied to parabolic equations, severe stability limits appear for the time step magnitude making the explicit simulations prohibitively expensive. Implicit schemes remove the stability limit, but each time step requires now the solution of linear (at best) or even nonlinear systems of equations. However, there exists a number of tricks to overcome (or at least to relax) severe stability limitations of explicit schemes without going into the trouble of fully implicit ones. The purpose of this manuscript is just to inform the readers about these alternative techniques to extend the stability limits. It was not written for classical scientific publication purposes.

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“…Following this topic the authors proposed an approach to porous medium flow modeling based on the analogy with the quasigasdynamic (QGD) system of equations [2]: the mass balance equation for each fluid phase was modified to get a regularizing term and to change the equation type from parabolic to hyperbolic [3]. Note that hyperbolization is a modern trend in CFD [4][5][6] to increase the stability of employed explicit schemes. The QGD-based model includes also the total energy conservation equation [7,8].…”

Section: Introductionmentioning

confidence: 99%

Non-Isothermal Compositional Model for Simulation of Multiphase Porous Media Flows

2019

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The research carried out in this work is aimed at the development of an explicit computational algorithm to simulate non-isothermal flow of multiphase multicomponent slightly compressible fluid through porous media. Such a modeling is necessary, for example, at solving practically important problems of hydrocarbon recovery via thermal methods and contaminated soil remediation via the hot steam injection. The original model and algorithm developed by the authors earlier draw the analogy with the quasi-gas-dynamic set of equations. Phase continuity equations were modified to get regularizing terms, while the type of equations was changed from parabolic to hyperbolic for the approximation by an explicit scheme with a mild enough stability condition. Since an adequate description of temperature-dependent porous media processes requires an explicit consideration of the transfer of mass and energy between the phases, the present work focuses on the generalization of the proposed approach to the case of multicomponent composition of fluids. Conservation laws are currently formulated for the components in terms of the mass concentrations of components in the phases. Constants of the phase equilibrium were used to close the set of equations. The developed model and algorithm have been verified numerically via test predictions of twoand three-phase flows, and, the phase transition of water into the gas phase at heating has been reproduced correctly.

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Estimates of the hyperbolization effect on the heat equation

Abstract: The difference between the solutions of the heat equation and its hyperbolized version is estimated. The estimates are obtained in the L 2 norm for the anisotropic heat equation and in the C norm for the one dimensional case with constant coefficients.

Cited by 20 publications

References 8 publications

Stable explicit schemes for simulation of nonlinear moisture transfer in porous materials

Stable explicit schemes for simulation of nonlinear moisture transfer in porous materials

How to overcome the Courant-Friedrichs-Lewy condition of explicit discretizations?

Non-Isothermal Compositional Model for Simulation of Multiphase Porous Media Flows

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