Convener(s)
| Name: | Dr. Samala Rathan | G.D. Veerappa Gowda |
| Mailing Address: | Assistant Professor, Faculty of Mathematics Department of Humanities & Sciences Indian Institute of Petroleum and Energy, Visakhapatnam-530003 |
Raja Ramanna Fellow TIFR Centre for Applicable Mathematics Sharadanagar, Chikkabommasandra, Yelahanka New Town, Bangalore - 65. |
| Email: | rathans.math at iipe.ac.in | gowda at tifrbng.res.in |
The workshop is on numerical methods for ordinary and partial differential equations (ODEs & PDEs) like those arising in science and engineering applications, such as wave, heat and sound propagation, fluid flow, chemical kinetics, etc. Generally, the solutions of differential equations are quite challenging, admit to limited regularity, may have complex structures in their form, and are highly nonlinear. It may not be possible to obtain an analytical solution even if we know the theoretical existence and uniqueness of the solution. Numerical methods offer a way to discretize the differential equations and then solve them. Thus, this workshop introduces some important numerical methods for linear and non-linear ordinary and partial differential equations and various numerical discretization techniques for their solution. We mainly focus on the construction of finite difference (FDM), finite volume methods (FVM), and discontinuous Galerkin methods (DGM) and their numerical convergence for the various nonlinear ODEs & PDEs. Higher-order methods are essential for wave propagation phenomena. Here, we discuss modern methods like WENO and DG for various applications of wave propagation such as hyperbolic conservation laws, Hamilton-Jacobi equation, and Convection-Diffusion-Dispersive type nonlinear PDEs. Machine learning and Deep learning techniques are quite popular nowadays, thus this workshop intends to introduce Machine Learning, Deep Learning, and Physics Informed Neural Networks (PINNs) and their applications to solve partial differential equations. This workshop is intended for PhDs and Postdocs so that they can learn these techniques and use them in their research.
Dates:
Venue:
Venue Address:
Dr. Samala Rathan
Assistant Professor, Faculty of Mathematics
Department of Humanities & Sciences
Indian Institute of Petroleum and Energy, Visakhapatnam-530003
Venue State:
Venue City:
PIN:
Chrono Order:
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.
Selected Applicants:
All the shortlisted applicants have been requested to confirm their participation through email to rathans.math@iipe.ac.in by 24-10-2024, failing which their selection will stand canceled.
| S.NO. | SID | Full Name | Gender | Affiliation | Position in College/University | University/Institute M.Sc./M.A. | Year of Passing M.Sc./M.A | Ph.D. Degree Date |
| 1 | 58936 | Mr Subhendu Mohanty | Male | Dhirubhai Ambani Institute of Information and Communication Technology, Gandhinagar | PhD Student | Central University of Kerala | 2020 | |
| 2 | 58959 | Ms Akanksha Singh | Female | National Institute of Technology Rourkela | PhD | Guru Ghasidas Vishwavidyalaya, Bilaspur (C.G.) India | 2018 | |
| 3 | 58983 | Mrs. Neha Kumari | Female | National Institute of Technology, Patna | PhD student | Ranchi University | 2017 | |
| 4 | 59002 | Mr. Rathindra Nath Basak | Male | Visva-Bharati (A Central University) | PhD | M.Sc from Tezpur University | 2020 | |
| 5 | 59044 | Miss Anju Anju | Female | Visvesvaraya National Institute of Technology | Junior Research Fellow | IIT Mandi/MSc | 2021 | |
| 6 | 59136 | Mr Harish Mallesham Nagula | Male | Institute of Chemical Technology - Indian Oil Campus, Bhubaneshwar. | PhD | M.Sc. | 2018 | |
| 7 | 59172 | Ms. Nikshita Pandya | Female | Central University Of Rajasthan,Ajmer | Ph.D. | The Maharaja Sayajirao University Of Baroda | 2022 | |
| 8 | 59179 | Mr. Simanchal Sahu | Male | Visvesvaraya National Institute of Technology | PhD | Berhampur University | 2021 | |
| 9 | 59196 | Mr. Vivek R | Male | CEG Campus, Anna University | PhD | Madras Christian College (Autonomous) | 2016 | |
| 10 | 59273 | Ms. Sarita Sihag | Female | BITS Pilani KK Birla Goa campus | PhD | National Institute of Technology Silchar Assam | 2022 | |
| 11 | 59282 | Mr. Daniel Vijay L | Male | IIT Goa | PhD | Bharathidasan University Tiruchirappalli | 2020 | |
| 12 | 59323 | Ms. Vaishnavi G | Female | Central University of Tamil Nadu | PhD | Central University of Tamil Nadu | 2021 | |
| 13 | 59347 | Mr. Kumar Mritunjay | Male | Rajiv Gandhi Institute of Petroleum Technology | Junior research fellow | Central University of Haryana | 2021 | |
| 14 | 59381 | Mr Muhammad Murtaza Tantry | Male | Vellore Institute of Technology | PhD | University of Hyderabad | 2020 | |
| 15 | 59451 | Mr. Zaffar Mehdi Dar | Male | Vellore Institute of Technology-Vellore | PhD | Jamia Millia Islamia | 2021 | |
| 16 | 59551 | Ms. Heena | Female | National Institute of Technology Warangal | PhD | Banaras Hindu University | 2019 | |
| 17 | 59583 | Ms. Susmita Saha | Female | University of Calcutta | PhD Student | University of Calcutta | 2018 | |
| 18 | 59644 | Mr Shubham Upadhyay | Male | Mahindra University | PhD | University Of Allahabad | 2022 | |
| 19 | 59682 | Ms Sarita Panda | Female | Mahindra University | PhD Scholar | Ravenshaw University, Cuttack | 2021 | |
| 20 | 59686 | Mr Dhayalan G | Male | National Institute of Technology, Tiruchirappalli | PhD | Bharathiar University, Coimbatore | 2020 | |
| 21 | 59712 | Mr. Shiv Mishra | Male | Indian Institute of Technology Roorkee | PhD | Banaras Hindu University | 2022 | |
| 22 | 59763 | Mr. Punit Yadav | Male | Indian Institute of technology Delhi | PhD | Indian Institute of technology Kanpur | 2024 | |
| 23 | 59764 | Mr Nitin Kumar | Male | Indian Institute of Technology Delhi | Phd | Indian Institute of Technology kanpur | 2024 | |
| 24 | 59773 | Mr Amit Mainan | Male | University of Delhi | Ph.D. | Banaras Hindu University | 2020 | |
| 25 | 59793 | Mr. Muniyasamy M | Male | National Institute of Technology Karnataka | PhD Student | Pondicherry University | 2019 | |
| 26 | 59799 | Mr Rovan Jonathan Vaz | Male | BITS Pilani Goa Campus | PhD | Dept. Of Mathematics, University of Pune | 2014 | |
| 27 | 59848 | Mr. Prakash Singh | Male | Indian Institute of Technology (ISM) Dhanbad | PhD | National Institute of Technology Calicut, Kerala | 2021 | |
| 28 | 59852 | Mr. Lilu Sahu | Male | IIT Kharagpur | PhD | Banaras Hindu University, M.Sc. | 2022 | |
| 29 | 59920 | Mr. Sujoy Basak | Male | Indian Institute of Technology Delhi | PhD | Tezpur University | 2022 | |
| 30 | 59997 | Mr Bikky Kumar | Male | IIT Ropar | PHD | Indian Institute of Technology Delhi ,M.Sc | 2022 | |
| 31 | 60038 | Mr. Kuruku Ankith | Male | Central University of Karnataka | PhD | Pondicherry University | 2021 | |
| 32 | 60088 | Ms. Shilpa Dey | Female | Indian Institute of Technology Madras | PhD | National Institute of Technology Rourkela , M.Sc. | 2022 | |
| 33 | 60215 | Mr. Devansh Paresh Sonigra | Male | Tata Institute of Fundamental Research- Centre for Applicable Mathematics | MSc Student | Tata Institute of Fundamental Research- Centre for Applicable Mathematics M.Sc | ||
| 34 | 60271 | Mr. Shuvam Banerjee | Male | National Institute of Science Education and Research | Integrated MSc-PhD student | National Institute of Science Education and Research, Bhubaneswar | Appeared / Awaiting Result | |
| 35 | 59170 | Mrs Anuradha Sahu | Female | SRM Institute of Science and Technology | phd | Berhampur University Berhampur Odisha | 2019 |
How to Reach:
TBA