I was invited to give a series of five one-hour lectures on numerical methods for quantum graphs at a Summer School at Université Polytechnique Hauts-de-France in June, 2024. Thanks to the organizers
- Colette De Coster (UPHF, Valenciennes)
- Damien Galant (UPHF, Valenciennes and UMONS, Mons, Belgium)
- Louis Jeanjean (Université de Franche-Comté, Besançon)
- Stefan Le Coz (Université Paul Sabatier, Toulouse)
Below are my handwritten lecture notes and links to my supplementary slides with numerical examples.
Lecture Notes
- Lecture 1: Finite-difference discretization of the Poisson problem on a quantum graph.
- Handwritten notes
- Accompanying slides (click play button to see builds and animations)
- Lecture 2: Rectangular Chebyshev collocation for the Poisson problem on a quantum graph.
- Lecture 3: Application of the framework to other numerical problems
- Lecture 4: Implicit-explicit time stepping
- Lecture 5: Continuation methods
Computational Examples
Examples from Delio Mugnolo’s lectures
- Example 1 An upper bound on the eigevalues of a quantum graph Laplacian.
- Example 2 A lower bound on the eigevalues of a quantum graph Laplacian.
- Example 3 Monotonicity of the eigevalues of a quantum graph Laplacian as the length of an edge is increased.
Examples from Diego Noja’s lectures
- Orbital Stability Example Considers two solutions to NLS on a dumbbell graph, one orbitally stable and the other unstable. Demonstrates what these look like in numerical simulations.
References
Main reference
- Goodman, Roy H, Grace Conte, and Jeremy L Marzuola. 2023. QGLAB: A MATLAB Package for Computations on Quantum Graphs. arXiv 2401.00561.
Inspirations for this work
Besse, Christophe, Romain Duboscq, and Stefan Le Coz. 2021. “Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library.” arXiv 2103.09650.
Goodman, Roy H. 2019. “NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph.” Discrete & Continuous Dynamical Systems - A 39: 2203–32.
Kairzhan, Adilbek, Dmitry E Pelinovsky, and Roy H Goodman. 2019. “Drift of Spectrally Stable Shifted States on Star Graphs.” SIAM Journal on Applied Dynamical Systems 18: 1723–55.
Marzuola, Jeremy L, and Dmitry E Pelinovsky. 2016. “Ground State on the Dumbbell Graph.” Applied Mathematics Research eXpress 2016: 98–145.
Spectral methods for boundary value problems
Aurentz, Jared L., and Lloyd N. Trefethen. 2017. “Block Operators and Spectral Discretizations.” SIAM Review 59: 423–46.
Boyd, John P. 2000. Chebyshev and Fourier Spectral Methods. Mineola, NY: Dover Publications.
Driscoll, Tobin A, and Nicholas Hale. 2015. “Rectangular spectral collocation.” IMA Journal of Numerical Analysis 38: 108–32.
Trefethen, Lloyd N. 2000. Spectral Methods in Matlab. SIAM.
Xu, Kuan, and Nicholas Hale. 2016. “Explicit construction of rectangular differentiation matrices.” IMA Journal of Numerical Analysis 36: 618 - 632.
Implicit-explict Runge-Kutta methods
- Ascher, Uri M, Steven J Ruuth, and Raymond J Spiteri. 1997. “Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations.” Applied Numerical Mathematics 25: 151–67.
Continuation Methods
Dhooge, A, Willy J F Govaerts, Yu A Kuznetsov, H G E Meijer, and B Sautois. 2008. “New features of the software MatCont for bifurcation analysis of dynamical systems.” Mathematical and Computer Modelling of Dynamical Systems 14: 147–75.
Dhooge, Annick, Willy J F Govaerts, and Yu A Kuznetsov. 2003. “MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs.” ACM Transactions on Mathematical Software 29: 141–64.
Doedel, Eusebius, and Bart Oldemann. “AUTO-07P : Users Manual.”
Nayfeh, Ali Hassan, and Balakumar Balachandran. 1995. Applied nonlinear dynamics: analytical, computational and experimental methods. Weinheim: Wiley VCH.