Solution of a One-Dimension Heat Equation Using Higher-Order Finite Difference Methods and Their Stability

Ali, Md. Emran; Loskor, Wahida Zaman; Taher, Samia; Bilkis, Farjana

doi:10.4236/jamp.2022.103060

Cited by 2 publications

(1 citation statement)

References 7 publications

(17 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Concerning the numerical solution of the heat equation, the usual methods are the finite difference methods (see [1] [2] [3]), the methods of lines (see [4] [5] [6]) and more recently the space-time finite element methods (FEMs) (see [7]- [12]).…”

Section: Introductionmentioning

confidence: 99%

A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities

Nkeck

2024

JAMP

View full text Add to dashboard Cite

The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.

show abstract

Section: Introductionmentioning

confidence: 99%

A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities

Nkeck

2024

JAMP

View full text Add to dashboard Cite

show abstract

An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation

Pei,

Jian,

2023

Applied Sciences

View full text Add to dashboard Cite

The stability and accuracy of explicit high-order finite difference (HOFD) algorithms have been research hotspots in different fields. To improve the stability and accuracy of the HOFD algorithms in thermal simulations, we present a Lax-Wendroff high-order finite difference (LHOFD) algorithm to solve the 2D transient heat transfer equation in this paper and develop an improved LHOFD (IHOFD) algorithm to improve the stability of the LHOFD algorithm. The formulas of the general high-order central FD (HOCFD) coefficients and the truncation error coefficient as well as the high-order non-central FD (HONFD) coefficients and the truncation error coefficient of the fourth-order spatial derivative are derived concisely in a different way. Furthermore, a unified analytical formula of the general HOCFD and HONFD coefficients, which can calculate the spatial derivative of any integer order, is derived. A new strategy of combination with the HOCFD and HONFD approximations under the same high-order accuracy as the internal computational domain is proposed to calculate the mixed derivatives of the boundary domains with high accuracy, no additional computational cost, and easy implementation. Then, the accuracy analysis, stability analysis, and comparative analysis of numerical simulation results obtained by the LHOFD and IHOFD algorithms with the exact solution show the correctness and validity of the proposed algorithms and their stability formulas, and the advantages of the proposed algorithms. The proposed algorithms are valid under both symmetric and asymmetric boundary conditions. The stability factor of the LHOFD algorithm is slightly higher than that of the conventional algorithm. The stability factor of the IHOFD algorithm is twice that of the conventional algorithm, and the maximum absolute error of the thermal simulation is within 0.015 (°C).

show abstract

Solution of a One-Dimension Heat Equation Using Higher-Order Finite Difference Methods and Their Stability

Cited by 2 publications

References 7 publications

A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities

A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities

An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation

Contact Info

Product

Resources

About