Abstract
The stability properties of a numerical method for the dual-phase-lag (DPL) equation are analyzed. The DPL equation has been increasingly used to model micro- and nanoscale heat conduction in engineering and bioheat transfer problems. A discretization method for the DPL equation that could yield efficient numerical solutions of 3D problems has been previously proposed, but its stability properties were only suggested by numerical experiments. In this work, the amplification matrix of the method is analyzed, and it is shown that its powers are uniformly bounded. As a result, the unconditional stability of the method is established.
Highlights
Non-Fourier heat conduction models have been increasingly used in recent years to model a variety of engineering and biological heat transfer problems
The numerical solution procedure for the dual-phase-lag model (DPL) equation proposed by McDonough et al [18] consists in applying first trapezoidal integration to (2) and using the following finite difference approximations: (
The use of non-Fourier models of heat conduction in engineering problems requires the use of efficient methods to compute numerical solutions, and a basic condition for these numerical methods to be reliably employed is to be confident that they present good stability properties
Summary
Non-Fourier heat conduction models have been increasingly used in recent years to model a variety of engineering and biological heat transfer problems (see, e.g., [1,2,3] and references therein). The aim of this note was to prove the unconditional stability of the finite difference method proposed in [18], which will be accomplished by analyzing its amplification matrix (e.g., [19, 20]). This type of von Neumann or Fourier stability analysis is directly valid for pure initial-value problems and for initial-boundary-value problems with periodic boundary conditions, and it can be applied to general Dirichlet or Neumann boundary conditions by using appropriate periodic extensions, provided that consistent discretizations are used for the boundary conditions [21, Ch. 3].
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