NCMW - Numerical Methods for Partial Differential Equations (2022)

Speakers and Syllabus


Name of the Speaker with affiliation No.of Lectures Detailed Syllabus
Veerappa Gowda G. D., Raja Ramanna Fellow, TIFR Centre for Ap- plicable Mathematics, Bengaluru 4 lectures of 1 and half
hours each
Non-linear hyperbolic conservation laws:
Con- cept of weak solutions, entropy condition and uniqueness of the solution. Construction of Godunov, Lax-Friedrichs, Engquist-Osher and Lax-Wendroff schemes. Lax-Wendroff theorem for convergence of the numerical solution. Fi- nite volume schemes. Monotone schemes: L∞ stability, total variation diminishing property and their convergence to the entropy solution. Higher order schemes. Finite volume schemes in higher dimensions.
Praveen Chandrashekar,
Associate Professor, TIFR-Centre for Ap- plicable Mathematics, Bengaluru
4 lectures of 1 and half
hours each
Discontinuous Galerkin method for hyperbolic PDE:
The lectures will provide an introduction to DG methods starting with 1-D problems. The topics covered include: numerical flux, semi- discrete stability, basis functions, implementa- tion details, limiters and TVD property, and, Fourier stability analysis. Then we will discuss DG for hyperbolic systems, multi-dimensional scheme, modal and nodal versions, special ver- sions like flux reconstruction, etc. The lectures will be complemented with examples of com- puter codes implementing these methods.
Sashi Kumar Ganesan,
Associate Professor, Com- putational Data Science, IISc Bengaluru
4 lectures of 1 and half
hours  each
Introduction to Computational Science,
Banach and Hilbert spaces, Weak derivatives, Sobolev spaces, A general elliptic problem of second or- der and its weak solution, Standard Galerkin method, Abstract error estimate. Implementa- tion of the finite element method: Mesh han- dling and data structure, Numerical integration, Sparse matrix storage, Assembling of system matrices and load vectors, Inclusion of bound- ary conditions. Introduction to Machine Learn- ing, Deep Learning. Introduction to Physics In- formed Neural Networks (PINNs).
K.R.Arun,
Assistant Professor, School of Mathematics, IISER Thiruvananthapuram
2 lectures of 1 and half
hours  each
IVP for ODEs :
One step methods: consis- tency and convergence; Runge-Kutta methods; Linear- Multistep methods: zero stability and consistency; Stiff equations and absolute stabil- ity
Sudarshan Kumar K,
Assistant Professor, School of Mathematics, IISER Thiruvananthapuram
2 lectures of 1 and half
hours  each
Linear PDEs:
Linear advection equation: fi- nite difference scheme, consistency and stability; Heat equation: finite difference schemes, theta methods, stability; Poisson equation: finite dif- ference scheme; Wave equation in 1D: L2 stabil- ity.

 

 


Time Table

Time/Day 19-09-2022 20-09-2022 21-09-2022 22-09-2022 23-09-2022 24-09-2022
09:00-10:30 KRA GDV PCR SKG SKG PCR
10:30-11:00 Tea Tea Tea Tea Tea Tea
11:00-12:30 SKK SKK SKG PCR PCR GDV
12:30-14:00 Lunch Lunch Lunch Lunch Lunch Lunch
14:00-15:30 KRA GDV T(SKG+TAH)
(one hour)
SKG GDV T(PCR+RKR)
(one hour)
15:30-16:00 Tea Tea Tea Tea Tea Tea
16:00-17:00 T(SKK+AVS) T(GDV+AVS)   T(SKG+TAH) T(PCR+RKR)  

 

 
• Full forms for the abbreviations of speakers and tutors:

Speakers
– SKK: Sudarshan Kumar K (Assistant Professor, School of Mathaematics, IISER Thiruvananthapuram)
– KRA: K.R. Arun (Assistant Professor, School of Mathaematics, IISER Thiruvananthapuram)
– GDV: G D Veerappa Gowda (Professor, TIFR-CAM, Bangalore)
– PCR: Praveen Chandrashekar (Associate Professor, TIFR-CAM, Bangalore)
– SKG: Sashi Kumar Ganesan (Associate Professor, Computational Data Science, IISc Bengaluru)
Tutors
– RKR: Rakesh Kumar (Postdoc, School of Mathaematics, IISER Thiruvananthapuram)
– TAH: Thivin Anandh (SRF, Computational Data Science, IISc Bengaluru)
– AVS: Aswin V.S (Postdoc, School of Mathaematics, IISER Thiruvananthapuram)
References
– E. Süli & David F. Mayers, An Introduction to Numerical Analysis, Cambridge University Press(2003).
– J.C. Strikwerda, Finite difference schemes and Partial differential equations, Wordsworth and Brooks(1989).
– E. Godlewski and P.A. Raviart, Hyperbolic Systems of Conservation Laws, Ellipses(1991).
– Sashikumaar Ganesan & Lutz Tobiska, Finite elements: Theory and Algorithms, Cambridge University Press(2017).
– P.G. Ciarlet, The finite element method for elliptic problems, North-Holland(1978)
 

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