NCMW - Numerical Methods for Partial Differential Equations (2024)
Speakers and Syllabus
Name of the Speaker with affiliation |
No.of Lectures |
Detailed Syllabus |
Veerappa Gowda G. D., Raja Ramanna Fellow, TIFR Centre for Applicable Mathematics, Bengaluru |
5 lectures of 1 and half hrs each |
Non-linear hyperbolic conservation laws: Concept of weak solutions, entropy condition and uniqueness of the solution. Construction of Go- dunov, Lax-Friedrichs, Engquist-Osher and Lax-Wendroff schemes. Lax- Wendroff theorem for convergence of the numerical solution. Finite volume schemes. Monotone schemes: L1 stability, total variation diminishing prop- erty and their convergence to the entropy solution. Higher order schemes. Finite volume schemes in higher dimensions. |
Praveen Chandrasekhar, Associate Professor, TIFR-Centre for Ap- plicable Mathematics, Bengaluru |
5 lectures of 1 and half hrs each |
Discontinuous Galerkin method for PDEs: The lectures will provide an introduction to DG methods starting with 1-D problems. The topics cov- ered include: numerical flux, semi-discrete stability, basis functions, imple- mentation details, limiters and TVD property, and, Fourier stability analy- sis. Then we will discuss DG for hyperbolic systems, multi-dimensional scheme, modal and nodal versions, special versions like flux reconstruc- tion, etc. The lectures will be complemented with examples of computer codes implementing these methods. |
Harish Kumar, Associate Professor, Department of Mathematics, IIT Delhi |
5 lectures of 1 and half hrs each |
Entropy stable schemes: Entropy inequality for the conservation laws. Fi- nite volume methods and entropy stability. Tadmor’s condition for entropy conservation. Entropy conservative fluxes, Entropy diffusion operator, En- tropy scaled right eigenvector. Semi-discrete Entropy stable finite differ- ence scheme. Discontinuous Galerkin schemes: Summation-by-parts prop- erty. Entropy conservative and entropy stable DG methods. Source terms and entropy stability. |
Sudarshan Kumar K, Assistant Professor, School of Mathemat- ics, IISER Thiruvanan- thapuram |
5 lectures of 1 and half hrs each |
Numerical methods for Ordinary Differential equations(ODEs): Initial value problem for ODEs: One step methods: consistency and convergence, Runge-Kutta methods, Linear multistep methods, zero stability and consis- tency, Stiff equations and absolute stability. Numerical methods for Linear PDEs: Linear advection equation: finite difference scheme, consistency, Von-Neumann stability, stability and convergence, Lax-equivalence theorem, Heat equation: finite difference schemes, theta methods, stability, Poisson equation: finite difference scheme, Wave equation in 1D: L2 stability. |
Samala Rathan, Assistant Professor, Department of Humani- ties and Sciences, IIPE Visakhapatnam |
5 lectures of 1 and half hrs each |
Introduction to higher order schemes: Lax-Wendorff scheme, flux-limiters and slope limiters, TVD schemes, MUSCL scheme, Central schemes; Semi-discrete formulations, WENO schemes: Polynomial and nonpoly- nomial basis reconstructions; Analysis of critical points, discontinuities and convergence properties. Applications of WENO schemes to Hamilton- Jacobi and convection-diffusion-dispersive type nonlinear PDEs. The lec- tures will be complemented with examples of computer codes implement- ing these methods. |
Ritesh Kumar Dubey, Associate Professor, |
5 lectures of 1 and half hrs each |
1. Deep Learning Introduction: Introduction to Deep Learning, Multilayer perceptrons, initialization strategies, important hyperparameters, optimization algorithms, activation functions, and regularization techniques.Deep Learning Algorithms: Supervised, Unsupervised, and Reinforcement Learning. 2. Deep Learning Architectures: Convolutional neural networks, Residual Networks, etc. 3. Deep Learning for Solving Differential Equations: Physics-informed neural networks (PINNs) and applications to solve forward and inverse ODE/PDE problems, Data driven and deep learning approach to solve hyperbolic conservation laws and related PDE’s, DeepOnet or Neural ODE’s. |
Time Table
Week1
Time/Day |
16-12-2024 |
17-12-2024 |
18-12-2024 |
19-12-2024 |
20-12-2024 |
09:30-11:00 |
SKK |
GDV |
RKD |
SKK |
RKD |
11:00-11:30 |
Tea |
Tea |
Tea |
Tea |
Tea |
11:30-1:00 |
GDV |
SKK |
GDV |
RKD |
SKK |
1:00-2:00 |
Lunch |
Lunch |
Lunch |
Lunch |
Lunch |
2:00-3:30 |
SKK |
GDV |
RKD |
GDV |
RKD |
3:30-4:00 |
Tea |
Tea |
Tea |
Tea |
Tea |
4:00-5:00 |
T(SS/NK/SU) |
T(SS/NK/SU) |
T(SS/NK/SU) |
T(SS/NK/SU) |
T(SS/NK/SU) |
Week 2
Time/Day |
23-12-2024 |
24-12-2024 |
25-12-2024 |
26-12-2024 |
27-12-2024 |
09:30-11:00 |
HK |
PC |
SR |
HK |
SR |
11:00-11:30 |
Tea |
Tea |
Tea |
Tea |
Tea |
11:30-13:00 |
PC |
HK |
PC |
SR |
HK |
1:00-2:00 |
Lunch |
Lunch |
Lunch |
Lunch |
Lunch |
2:00-3:30 |
HK |
PC |
SR |
PC |
SR |
3:30-4:00 |
Tea |
Tea |
Tea |
Tea |
Tea |
4:00-5:00 |
T(AB/LVS/RK) |
T(AB/LVS/RK) |
T(AB/LVS/RK) |
T(AB/LVS/RK) |
T(AB/LVS/RK) |
◦ Full forms for the abbreviations of speakers:
▪ SKK: Sudarshan Kumar K (IISER, Thiruvananthapuram)
▪ GDV: G D Veerappa Gowda (TIFR-CAM, Bangalore)
▪ PC: Praveen Chandrashekar (TIFR-CAM, Bangalore)
▪ HK: Harish Kumar (IIT, Delhi)
▪ RKD: Ritesh Kumar Dubey (SRMIST, Chennai )
▪ SR: Samala Rathan (IIPE, Visakhapatnam)
◦ Full forms for the abbreviations of tutors (if any):
▪ LVS: Lavanya V Salian (IIPE, Visakhapatnam)
▪ SS: Sanjibanee Sudha (IIPE, Visakhapatnam)
▪ SU: Sudipta Sahu (IIPE, Visakhapatnam)
▪ NK: Nikhil Manoj (IISER, Thiruvananthapuram)
▪ AB: Aadi Bhure (TIFR CAM Bangalore)
▪ RK: Rakesh Kumar (IISER, Thiruvananthapuram)
◦ References:
- L.C. Evans, Partial Differetial Equations, Graduate Studies in Mathematics, Vol 19, AMS 1998.
- E. Godlewski, P.A. Raviart, Hyperbolic systems of conservation laws, Mathematiques et Applications, Ellipses, Paris, 1991.
- E. Godlewski, P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Springer, 1996.
- R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser Verlag, 1992.
- R.J. LeVeque, Finite-Volume methods for hyperbolic problems, Cambridge Univ. Press, 2004.
- J.W. Thomas, Numerical Partial Differential Equations: conservation laws and elliptic equations, Springer, 2013.
- J.S. Hestheven, Numerical methods for conservation laws: From Analysis to Algorithms, SIAM pub- lisher, 2018.
- C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced numerical approximation of nonlinear hyperbolic equations(Lecture notes in mathematics), Berlin: Springer-Verlag; 1998. pp. 325-432 .
- G.S. Jiang, D. Peng: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)
- Y. Liu , C.W. Shu, M. Zhang, High order finite difference WENO schemes for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput. 33 (2011) 939–965.