On the Rayleigh-Bénard-Marangoni problem: Theoretical and numerical analysis
June 2020, 7(1): 123-158. doi: 10.3934/jcd.2020005
J. P. Lessard 1, and J. D. Mireles James 2,,
Department of Math. and Stat., McGill University, Burnside Hall, Room 1119,805 Sherbrooke West, Montreal, QC, H3A 0B9, CANADA
Department of Mathematical Sciences, Florid Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA
* Corresponding author: J. D. Mireles James
Received June 2019 Revised January 2020 Published March 2020
Fund Project: The first author is supported by the NSERC. The second author was partially supported by NSF grant DMS - 1813501
This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by $ N \times N $ matrices for large enough $ N $. We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.
Keywords: Validated numerics, delay differential equations, unstable eigenvector count, Chebyshev series.
Mathematics Subject Classification: Primary: 65G20, 37B30; Secondary: 34K08, 34K20, 65M70.
Citation: J. P. Lessard, J. D. Mireles James. A functional analytic approach to validated numerics for eigenvalues of delay equations. Journal of Computational Dynamics, 2020, 7 (1) : 123-158. doi: 10.3934/jcd.2020005
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$ g(\bar z) $ (quality of the approximate zero) or the value of $ g'(\bar z) $ (slope of the tangent line). Changing any of these values effects the location of the envelope, and hence the outcome of the proof. Given the data $ g(\bar z) $, $ g'(\bar z) $, and the second derivative bound the role of $ p(r) $ is to determine if the data is good enough to imply the existence of a zero for explicit values of $ r $. The actual proof of the theorem involves changing coordinates to flatten out the function over its tangent line and applying the contraction mapping theorem, and this introduces factors of $ a = 1/g(\bar z) $ throughout the hypotheses. In the proof the "envelope" is the size of the neighborhood on which the resulting map is a contraction">Figure 1. Schematic illustration of the meaning of Theorem 2.2: The idea behind why a Newton-Kantorovich type theorem works is that we find a point $ \bar z $ where $ g(\bar z) $ is small and evaluate the derivative $ g'(\bar z) $. If $ g'(z) \neq 0 $ then the tangent line (best linear approximation of $ g $ at $ \bar z $) must have a zero. Does it follow that $ g $ has a zero? The answer depends on the size of the second derivative, which lets us construct a confining "envelope" around the linear approximation. In the left frame we have a situation where the envelope forces the function to have a zero, and in the right frame the envelope is not tight enough and the function could "escape". Of course for a given bound on the second derivative (size of the envelope) changing the value of $ g(\bar z) $ (quality of the approximate zero) or the value of $ g'(\bar z) $ (slope of the tangent line). Changing any of these values effects the location of the envelope, and hence the outcome of the proof. Given the data $ g(\bar z) $, $ g'(\bar z) $, and the second derivative bound the role of $ p(r) $ is to determine if the data is good enough to imply the existence of a zero for explicit values of $ r $. The actual proof of the theorem involves changing coordinates to flatten out the function over its tangent line and applying the contraction mapping theorem, and this introduces factors of $ a = 1/g(\bar z) $ throughout the hypotheses. In the proof the "envelope" is the size of the neighborhood on which the resulting map is a contraction
$ \Gamma = \alpha + \beta + \gamma + \delta $ illustrated in the figure. If $ \tilde g $ is analytic inside the region enclosed by $ \Gamma $ then the argument principle counts the number of zeros inside. Supposing that there are no zeros or poles of $ \tilde g $ on $ [-1, 0) $, taking the limit as $ R \to 0 $ and $ d \to 0 $ gives the unstable eigenvalue count, i.e. the Morse index. By choosing other circles we could count the number of stable eigenvalue with modulus larger than some desired bound">Figure 2. Validated Morse index by contour integration: imagine that $ \lambda_1 $ and $ \lambda_2 $ zeros inside the unit circle for the function $ \tilde{g} $ defined in Equation (25), so that $ \lambda_1^{-1}, \lambda_2^{-1} $ are unstable eigenvalues for the step map $ F $. To validate the eigenvalue count we choose a branch of the complex logarithm defined on $ \mathbb{C} \backslash(-\infty, 0] $, and consider the "key hole" contour $ \Gamma = \alpha + \beta + \gamma + \delta $ illustrated in the figure. If $ \tilde g $ is analytic inside the region enclosed by $ \Gamma $ then the argument principle counts the number of zeros inside. Supposing that there are no zeros or poles of $ \tilde g $ on $ [-1, 0) $, taking the limit as $ R \to 0 $ and $ d \to 0 $ gives the unstable eigenvalue count, i.e. the Morse index. By choosing other circles we could count the number of stable eigenvalue with modulus larger than some desired bound
Figure 3. Representation of real analytic functions: Illustration of the complex extension of a real analytic function $ y(t) $ defined on $ [-\tau, 0] $ whose nearest complex pole is at $ z_0 \in \mathbb{C} $ with $ \mbox{dist}([-\tau, 0], z_0) < \tau $. There is a mesh $ \tau_0 = -\tau, \ldots, \tau_M = 0 $, and power series expansions $ y_0(t), \ldots, y_M(t) $ so that $ y(t) = y_j(t) $ for any $ t $ where the power series converges. Each power series $ y_j(t) $ is centered at the point $ t_j $ and converges on a disk of radius $ R_j = |t_j - z_0| $. We refer to these disks as $ C_j $, $ 0 \leq j \leq M $. The same function $ y(t) $ can be represented by a Chebyshev series converging absolutely and uniformly on the Bernstein ellipse $ E $ whose semi-minor axis is at least $ \rho = |\mbox{imag}(z_0)| $. The fact that one Chebyshev series is always sufficient to represent a real analytic function on $ [-\tau, 0] $ is a significant advantage for our discretization scheme
Figure 4. On the left, the spectrum computations for $ DF(u_0) $ and on the right, the spectrum computations for $ DF(u_1) $. The circles of radii $ r $ used in the computation of the generalized Morse indices in the Mackey-Glass equation (14) at the parameter values $ \tau = 2 $, $ \gamma = 1 $, $ \beta = 2 $ and $ \rho = 10 $ are plotted. On each plot, the unit circle is the largest one and is portrayed in red
Table 1. Parameters used in the proof of Theorem 4.4 to obtain the generalized Morse indices in the Mackey-Glass equation (14) at the parameter values $ \tau = 2 $, $ \gamma = 1 $, $ \beta = 2 $ and $ \rho = 10 $
$ u_0 $ $ r $ $ N $ $ \nu $ $ m $ $ \mu_r(u_0) $
0 1 32 1.3 10 1
0 0.6 70 1.2 10 3
0 0.29 200 1.15 40 5
0 0.2 600 1.05 60 7
1 1 310 1.1 100 2
1 0.85 130 1.2 30 4
1 0.341 500 1.05 90 8
Table 2. Parameters used in the proofs of the higher dimensional examples
$ d $ $ \tau $ $ N $ $ \nu $ $ m $ $ C_1 $ $ C_2 $ $ C_3 $ $ C $ $ \mu_1(u_0) $ Elapsed time (in ${\tt secs}$)
3 1 5 2.4 15 4.1174 4.0101 0.013124 0.21669 2 1.27
12 1 5 2.4 15 4.8091 4.9555 4.9555 0.4729 2 1.63
24 1 5 2.4 15 5.3788 5.9556 0.029061 0.9309 2 2.67
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J. P. Lessard J. D. Mireles James