A Numerical Method for Solving Nonlinear Heat Transfer Equations

Abstract

Heat transfer problems are usually governed by nonlinear differential equations, which, after discretization, result in a set of algebraic and transcendental equations with the nonlinearity retained. In the present study, a numerical method for solving such equations is proposed. The primary interest of the present study focuses on situations where the traditional Newton-Raphson method fails to converge.

The proposed method combines (1) the Newton-Raphson method, (2) the continuation method, and (3) perturbations of diagonal elements in the Jacobian matrices. When (3) is needed, it is possible to examine the magnitudes of diagonal elements, or those of eigenvalues of Jacobian matrices, for some guidance toward the choice of perturbations.

The Burgers' transient flow problem and a problem of transient two-dimensional heat conduction with nonlinear heat generation are solved to illustrate the proposed method. Some initial guesses led to situations in which a combination of all three methods must be used jointly to achieve successful convergence.

It should be emphatically noted that the convergence rates and accuracies are beyond the scope of the present study.

Notes

^(a) ξ = 0.76, ω(2) = 5.

^(b) ξ = 0, ω(2) = 5.

^(c) ξ = 0, ω(2) = 3.

^(d) ξ = 0, ω(2) = 0.

^(a) ξ = 0.00019, ω(1) = − 4, ω(2) = − 4.

^(b) ξ = 0.00019, ω(1) = 0, ω(2) = 0.

^(c) ξ = 0, ω(1) = 0, ω(2) = 0.