2.4 Quasi-unconditional stability for higher-order
2.4.1 Order-s BDF methods outside the region of quasi-
(a) (b) (c)
Figure 2.3: Demonstration of a CFL-like stability constraint when ∆t is outside the rectangular window of quasi-unconditional stability for the advection-diffusion equa- tion with α = 1.0 and β = 0.05 (parameters selected for clarity of visualization.
Theoretical value: Mt = 0.0965 for this selection of physical parameters). The eigen- values multiplied by ∆t (black dots) are plotted together with the boundary of the BDF5 stability region (dashed grey curve; cf. Figure 2.1). (a) Using N + 1 = 9 grid points and time step∆t= 0.23all eigenvalues lie within the stability region. (b) The number of points is increased toN+ 1 = 19while the time step is held constant. The ten additional eigenvalues are not in the stability region, which indicates the method is unstable for these parameter values. (c) The number of points is againN+ 1 = 19, but the time step is reduced to ∆t= 0.12, causing all eigenvalues to be contained in the stability region.
Theorems 2.4 and 2.5 should not be viewed as a suggestion that the s-th order BDF methods are not stable when the constraints in the theorem are not satisfied.
(a) (b) (c)
Figure 2.4: Continuation of Figure 2.3. (a) The time step is ∆t = 0.12 (as in Figure 2.3(c)) and the number of grid points is increased toN+ 1 = 35. Once again, some eigenvalues do not lie in the stability region. (b) The number of grid points is held at N + 1 = 35 while the time step is reduced to ∆t = Mt = 0.0965, which is the maximum allowed for the window of quasi-unconditional stability. All eigenvalues now lie in the stability region. (c) With ∆t= 0.0965, additional eigenvalues (arising from further increasing the number of grid points) remain within the stability region, thus demonstrating the quasi-unconditional stability of the BDF scheme of order 5.
Indeed, while, by definition, for ∆t > Mt the complete parabolic region Γm passes through the region where the BDF method is unstable (as demonstrated in Figure 2.3 as well as in the first two images in Figure 2.4), stability can still be ensured for such a value of∆t provided adequate values of the discretization parameterN+ 1 = 2π/h are used. Indeed, taking into account that only a bounded segment in the parabola is actually relevant to the stability of the ODE system that results for each fixed value of N, we see that stability may be ensured provided this particular segment, and not necessarily the complete parabolaΓm, is contained in the stability region of the s-th order BDF algorithm.
From equation (2.121) we see that increasing values of N lead to corresponding increases in the length of the parabolic segment on which the eigenvalues actually lie,
while decreasing∆t results in reductions of both the length of the relevant parabolic segment as well as the width of the parabola itself. Therefore, for∆t > Mt, increasing the number of grid points will inevitably cause some eigenvalues to eventually enter the region of instability. But stability can be restored by a corresponding reduction in ∆t—see Figure 2.3. This argument suggests that a CFL condition of the form
∆t ≤ C/N exists for ∆t > Mt. Of course, when ∆t is reduced to the value Mt or below, then no increases inN (reductions inh) result in instability—as demonstrated in Figure 2.4. We may thus emphasize: within the quasi-unconditional stability window no such CFL-like stability constraints exist.
Figure 2.5: Maximum stable ∆t versus spatial mesh size h for Fourier-based BDF and AB methods of orders three and four when applied to the advection-diffusion equation (2.115), with α= 1, β = 10−2 on the left andα = 1, β= 10−2 on the right.
To better understand when the BDF methods are preferable to an explicit scheme, we compare their stability to that of the explicit Adams-Bashforth (AB) multistep methods. For a given number of discretization points N + 1and physical parameter values α and β, we can use the equation for the eigenvalues (2.120) of the advection- diffusion equation and the boundary locus z(θ) of the stability regions to estimate the maximum stable ∆t by solvingz(θ) =λ(N/2) ∆t for θ and ∆t.
Figure 2.6: Maximum stable ∆t versus spatial mesh size h for Fourier-based BDF and AB methods of orders three and four when applied to the advection-diffusion equation (2.115), with α= 1, β = 10−3 on the left andα = 1, β= 10−4 on the right.
Figures 2.5 and 2.6 show the maximum stable ∆t allowed by the Fourier-based BDF and AB methods of orders three and four for the advection-diffusion equation, with various discretizations and values of the parameters α and β. We observe from the α = 1 plots that both the BDF and AB methods have a CFL-type constraint of the form∆t < Chfor large values ofh. Whenhis decreased to the order ofβthe CFL condition for the explicit method becomes more severe (∆t < Ch2). By this point, the BDF methods have already entered the of window quasi-unconditional stability.
At h = β, the stable ∆t for the BDF methods are about one hundred times larger than their AB counterparts. Clearly, the BDF methods are preferable in regimes where the AB methods suffer from the severe∆t < Ch2 CFL condition. However, as we will see in Tables 2.4, 2.5 and 2.6 in the next section, much higher stable ∆t can be achieved in practice by the BDF-ADI methods for the full Navier-Stokes equations in two dimensions than suggested by the linear stability analysis.