This class will cover the classical numerical analysis for PDEs. Particularly we will focus on high order numerical methods for PDEs such as spectral methods, WENO methods and Machine learning methods for PDEs. Spectral methods are high order methods based on global polynomials such as the trigonometric or orthogonal polynomials, and they yield the so called exponential or spectral convergence when smooth problems are considered. Due to such high order accuracy, spectral methods have been actively applied to various problems in applied mathematics. This course will introduce spectral methods with emphasis on both theory and applications. Spectral methods will be derived for various PDEs such as wave equations, heat equations, and nonlinear hyperbolic conservation laws. Recent developments of the spectral methods for non-smooth problems and the discontinuous Galerkin methods will be also covered. Students who take this course will have an understanding of spectral methods and be able to apply them to real computational problems. Besides spectral methods, some other high order methods will be also covered such as ENO, WENO methods and methods with neural network approach such as PINNs.

No prerequisites

• Homework Assignments 30%
• Midterm exam/Take home exam 30%
• Final project 30%
• Class Attendance/Presentation 10%

[1] Spectral methods for time-dependent problems by J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Cambridge UP, 2007.
[2] Advanced numerical approximation of nonlinear hyperbolic equations, Cockburn, Shu, Johnson, Tadmor, 1997, Springer – C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, https://link.springer.com/chapter/10.1007/BFb0096355
[3] DGM:A deep learning algorithm for solving partial differential equations, Sirignano & Spiliopoulos, https://arxiv.org/abs/1708.07469
[4] Can PINN beat the finite element method? Grossmann et al., https://arxiv.org/abs/2302.04107

Review: Hyperbolic conservation laws
Review: Polynomial interpolation
o Lagrange interpolation
o Runge phenomenon, Gibbs phenomenon
o Finite difference methods for ODEs & PDEs
Phase error analysis
Trigonometric polynomials
Fourier spectral methods
o Fourier-Galerkin methods
o Fourier-collocation methods
o Stability analysis of Fourier spectral methods
Orthogonal polynomials
o Polynomial spectral methods
o Chebyshev spectral methods/Legendre spectral methods
o Spectral-Galerkin/Collocation methods
o Stability analysis of spectral methods
Spectral methods for non-smooth problems
o The Gibbs phenomenon
o Spectral filtering methods
o Gegenbauer reconstruction methods
o Spectral edge detection methods
Penalty spectral methods
Computational aspects of spectral methods
Discontinuous Galerkin methods
Radial basis function methods
ENO-WENO methods for conservation laws
Physics-informed neural networks for PDEs

• Students are expected to attend the class regularly.
• Class participation and discussion among students are highly encouraged.
• Homework assignments should be submitted on time.
• Try to finish the final project either as a group or individually. To finish it successfully discuss on the progress with the instructor frequently during the semester.