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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.

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., [

Considering first-order approximations in the phase-lags in (

Analytical and numerical solutions for the DPL equation and other related DPL models derived from (

As pointed out in [

The aim of this note was to prove the unconditional stability of the finite difference method proposed in [

In the next section we recall the method proposed in [

The numerical solution procedure for the DPL equation proposed by McDonough et al. [

The resulting finite difference scheme is globally first-order accurate in time and second-order accurate in space. Assuming

We note the following bounds, which will be of use to obtain bounds on the eigenvalues of the amplification matrix. Since

Next we compute the eigenvalues of the amplification matrix, as written in (

Therefore, the following bound on the spectral radius of the amplification matrix holds:

We will next follow similar arguments to those used in [

Also, when there are two different real eigenvalues, it holds that

There are two especial cases of the DPL model that could be singled out. When

In this special case the method is a two-level scheme:

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.

The numerical solution procedure for the DPL equation proposed by McDonough et al. [

In this note, by analyzing the amplification matrix of the scheme, we have provided a detailed proof showing that the method is indeed unconditionally stable, including its possible application to the particular case when

As stressed in [

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was partially funded by Grant GRE12-08 from University of Alicante.