# 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)