Convergence of a Finite Difference Scheme for a Flame/Smoldering-Front Evolution Equation and Its Application to Wavenumber Selection

A Neutral Stability Curves

In this section, we describe a linearized stability analysis of (1.1) around the trivial solution u ⁢ ( σ , t ) ≡ 0 . Substituting the Fourier expansion u ⁢ ( σ , t ) = ∑ m ∈ ℤ u m ⁢ ( t ) ⁢ e - 1 ⁢ m ⁢ σ , u m ⁢ ( ⋅ ) ∈ ℂ into (1.1), we obtain an infinite-dimensional dynamical system

where λ ± 1 ≡ 0 and

Note that u - m ⁢ ( t ) = u ¯ m ⁢ ( t ) follows from u ⁢ ( ⋅ , ⋅ ) ∈ ℝ . By solving λ m = 0 on R, we obtain the neutral stability curves upon which the linearized operator of (1.1) has eigenvalues of zero. Note that λ ± 1 ≡ 0 holds for any R > 0 . However, λ m < 0 holds for each | m | = 2 , 3 , … when R < R * = 2 ⁢ δ α - 1 . Therefore, the circle solution is neutrally stable as indicated in the gray region in Figure 3. In the white region where R > R * , the circle is unstable except at the 2-mode neutral-stability curve because λ m > 0 holds for at least one integer m ∈ { ± 2 , ± 3 , … } . In particular, when | m | = 2 , for any fixed δ > 0 , the value of R * is the minimum value at which the stability of the circle solution changes from neutrally stable to unstable.