Harihara Maharna

Harihara Maharna

Applied Mathematics PhD Candidate at the University of Notre Dame

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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.

Pages

Posts

Future Blog Post

less than 1 minute read

Published:

This post will show up by default. To disable scheduling of future posts, edit config.yml and set future: false.

Blog Post number 4

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 3

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 2

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

Blog Post number 1

less than 1 minute read

Published:

This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

project

Differential Equations

Project, National Institute of Science Education and Research (NISER), Bhubaneswar, 2023

  • Guide: Dr. Anupam Pal Choudhury, School of Mathematics, NISER Bhubaneswar, India About the project: This work was done during my Summer Research Intern position at NISER Bhubaneswar.
  • In this project, I studied scalar conservation laws and how they model physical phenomena with a particular emphasis on traffic dynamics.
  • I learned about weak (or integral) solutions, Rankine-Hugoniot condition, and entropy conditions.

An Asymptotic Preserving and Energy Stable Scheme for the Euler System with Congestion Constraint

Project, Indian Institute of Science Education and Research (IISER), Thiruvananthapuram, Department of Mathematics, 2024

  • Guide: Dr. K. R. Arun, School of Mathematics, IISER Thiruvananthapuram, India
  • About the project: This work was conducted as part of my Master’s thesis at IISER Thiruvananthapuram.
    • In this project, we designed and analyzed a finite volume scheme for the barotropic Euler equations with the congestion pressure law and performed the singular limit termed as the hard congestion limit at the discrete level.
    • The developed scheme was an entropy stable and asymptotic preserving. We also obtained a-priori estimates on the relevant unknowns. We lastly, proved the efficiency of the numerical scheme by testing various numerical examples.

Impact of hemodynamic parameters on rupture risk in abdominal aortic aneurysm: Emphasis on wall shear stress-derived indicators

Class Project, University of Notre Dame, 2025

  • Course: ACMS 60792 Numerical hemodynamics and Uncertainty Quantifciation
  • Semester: Spring 2025
  • Instructor: Dr. Daniele E. Schiavazzi
  • Project Title: Impact of hemodynamic parameters on rupture risk in abdominal aortic aneurysm: Emphasis on wall shear stress-derived indicators
  • Investigated AAA Hemodynamics Through WSS-Derived parameters: This project focused on analyzing the role of wall shear stress (WSS) and its derived parameters, TAWSS, OSI, ECAP, and RRT, in the progression and rupture risk of abdominal aortic aneurysms (AAAs), enhancing understanding of disturbed blood flow patterns.
  • Utilized SimVascular for Computational Modeling: A representative AAA model and a virtually repaired version were studied using SimVascular to compute key hemodynamic metrics, offering insights into how arterial geometry influences shear stress and potential rupture sites.
  • Read the full project report (PDF).

Deep Learning for Complex Fluids

Project, University of Notre Dame, 2025

Advisor: Dr. Zhiliang Xu, Professor, ACMS Department, University of Notre Dame

This is my primary doctoral research, conducted as part of my Graduate Research Assistantship.

I am developing the displacement-based JKO scheme — a structure-preserving deep learning approach for solving complex fluids. The method is grounded in the Energetic Variational Approach (EnVarA), which derives dynamics from the interplay between energy and dissipation. By constructing the numerical scheme directly from the energy-dissipation law, the scheme guarantees monotonic decay of the system’s free energy, preventing unphysical states and ensuring long-term stability.

The current focus is the Cahn-Hilliard equation for phase-separation dynamics. Key contributions include:

  • A displacement-based neural network discretization that uses neural networks as mesh-free spatial discretizers, enabling scalability to high dimensions.
  • A gradient-based adaptive refinement strategy for better capturing diffuse interfaces.
  • An implicit midpoint symplectic time-stepping scheme with provable energy stability.

    The framework is implemented in PyTorch and is being extended to coupled Cahn-Hilliard–Navier-Stokes systems for modeling complex fluid dynamics.

publications

talks

Energetic Variational Neural Network Discretization of the Cahn-Hilliard Equation

Published:

In this talk, I presented a structure-preserving Lagrangian algorithm for solving the Cahn-Hilliard equation. The algorithm employs neural networks as tools for spatial discretization. The proposed scheme is constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the system’s free energy, which avoids unphysical states of solutions and is crucial for the long-term stability of numerical computations. To address challenges arising from interface problems, we introduce an adaptive sampling method for better capturing the diffuse-interface. Moreover, we solve for the incremental of the flow map. This approach is computationally memory-efficient. The proposed neural network-based scheme is mesh-free, allowing us to solve gradient flows in high dimensions. Numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical schemes.

teaching

Numerical Analysis (ACMS 40390)

Undergraduate course, University of Notre Dame, ACMS Department, 2025

As a Teaching Assistant, I held office hours and grade homework.