Convergency and Stability of Explicit and Implicit Schemes in the Simulation of the Heat Equation

Carreño, Franyelit María Suárez; Rosales-Romero, Luis

doi:10.3390/app11104468

Cited by 17 publications

(9 citation statements)

References 36 publications

(59 reference statements)

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“…First, their formula is given for the simplest case, i.e., one dimensional, equidistant mesh, Equation ( 4), which is used for verification. The more general forms, which can be applied to (6) are immediately given because these will be used to simulate the heat transfer in the wall.…”

Section: The Applied Numerical Methodsmentioning

confidence: 99%

“…Here the (t, x) ∈ [5,6] × [6, 10] computational domain is considered with the same parameters as in the previous subsection. The initial function has a very similar exponential curve as in Figure 3.…”

Section: Strong Nonlinearitymentioning

confidence: 99%

“…We are aware that plenty of numerical methods are used to solve heat transfer and similar problems, such as finite difference schemes (FDM) [5][6][7] and finite element methods (FEM) [8]. However, they can be extremely time-consuming since the examined system must be fully discretized both in space and time.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Comparison of Old and New Stable Explicit Methods for Heat Conduction, Convection, and Radiation in an Insulated Wall with Thermal Bridging

2022

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Using efficient methods to calculate heat transfer in building components is an important issue. In the current work, 14 numerical methods are examined to solve the heat transfer problem inside building walls. Not only heat conduction but convection and radiation are considered as well, in addition to heat generation. Five of the used methods are recently invented explicit algorithms, which are unconditionally stable for conduction problems. First, the algorithms are verified in a 1D case by comparing the results of the methods to an analytical solution. Then they are tested on real-life cases in the case of surface area (made of brick) and cross-sectional area (two-layer brick and insulator) walls with and without thermal bridging. Equidistant and non-equidistant grids are used as well. The goal was to determine how the errors depend on the properties of the materials, the mesh type, and the time step size. The results show that the best algorithms are typically the leapfrog-hopscotch and the modified Dufort–Frankel and odd–even hopscotch algorithms since they are quite accurate for larger time step sizes, even for 100 s as well.

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Section: The Applied Numerical Methodsmentioning

confidence: 99%

Section: Strong Nonlinearitymentioning

confidence: 99%

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Comparison of Old and New Stable Explicit Methods for Heat Conduction, Convection, and Radiation in an Insulated Wall with Thermal Bridging

2022

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“…Although it can be as small as the numerical methods and calculation resources allow, this error is always present, and its handling must be considered in developing the solutions. This research shows a different alternative to simple iterations to find solutions close to reality [4].…”

Section: Introductionmentioning

confidence: 98%

Simulations in Science and Engineering for University Education

Bejarano,

Bedregal

et al. 2023

ijmst

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In this paper, the strategies and tests that guarantee a reliable solution of partial differential equations (PDE) solved with the finite difference method are proposed. The elements to be considered are presented so that the solution of equations in partial derivatives is the closest to reality. The experience of the proponents and previous work play an important role in validating simulations in science and engineering problems. A series of tests are proposed that must be carried out to validate and make the simulation results reliable, due to the fact that the selection of the finite difference scheme, the initial data, the boundary conditions and the restrictions in the time step, the stability and convergence of the solution cannot be guaranteed a priori. There are situations in which the solution is stable but converges to values ??that have no physical meaning. To illustrate the strategies and tests inherent in a simulation, such that the solutions can be trusted, the 3D wave equation (three spatial dimensions and one temporal dimension) is solved numerically, by implementing a code in FORTRAN 90, implementing a scheme finite difference to second order approximation. The stability and convergence of the solution are studied according to the schemes proposed in this research, in such a way that the results are reliable, and guarantee the possible practical implementation of the solutions. Finally, the evolution of the scalar field for different times is shown, validated with the principle of conservation of energy. The strategies implemented in this work to guarantee the reliability of the simulations are also valid when solving EDP problems with neural networks, genetic algorithms and other intelligent computing techniques.

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“…There are lots of numerical methods to solve the heat conduction equation, such as several finite difference schemes (FDM) [5][6][7], finite element methods (FEM) [8], or a combination of these [9]. However, they can be computationally demanding since they require the full spatial discretization of the examined system.…”

Section: Introductionmentioning

confidence: 99%

A Comparative Study of Explicit and Stable Time Integration Schemes for Heat Conduction in an Insulated Wall

2022

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Calculating heat transfer in building components is an important and nontrivial task. Thus, in this work, we extensively examined 13 numerical methods to solve the linear heat conduction equation in building walls. Eight of the used methods are recently invented explicit algorithms which are unconditionally stable. First, we performed verification tests in a 2D case by comparing them to analytical solutions, using equidistant and non-equidistant grids. Then we tested them on real-life applications in the case of one-layer (brick) and two-layer (brick and insulator) walls to determine how the errors depend on the real properties of the materials, the mesh type, and the time step size. We applied space-dependent boundary conditions on the brick side and time-dependent boundary conditions on the insulation side. The results show that the best algorithm is usually the original odd-even hopscotch method for uniform cases and the leapfrog-hopscotch algorithm for non-uniform cases.

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Convergency and Stability of Explicit and Implicit Schemes in the Simulation of the Heat Equation

Cited by 17 publications

References 36 publications

Comparison of Old and New Stable Explicit Methods for Heat Conduction, Convection, and Radiation in an Insulated Wall with Thermal Bridging

Comparison of Old and New Stable Explicit Methods for Heat Conduction, Convection, and Radiation in an Insulated Wall with Thermal Bridging

Simulations in Science and Engineering for University Education

A Comparative Study of Explicit and Stable Time Integration Schemes for Heat Conduction in an Insulated Wall

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