Prof. Siva Nadarajah is a Professor in the Department of Mechanical Engineering at McGill University and Director of the McGill Institute for Aerospace Engineering (MIAE). He leads the Computational Aerodynamics Group, where his research focuses on the development of next-generation algorithms for aerodynamic design optimization, high-order computational fluid dynamics, large-eddy simulation, adaptive mesh refinement, reduced-order modelling, and high-performance computing. He is internationally recognized for pioneering contributions to adjoint-based aerodynamic shape optimization and high-order numerical methods for complex aerospace flows.
In recent years, Prof. Nadarajah’s research has advanced nonlinear stability theory for flux reconstruction and discontinuous Galerkin methods, leading to some of the first provably stable high-order schemes for compressible flows. His recent publications have focused on entropy-stable discretizations, robust shock-capturing strategies, wall-modelled large-eddy simulation, and adaptive high-order methods for turbulent flows. He has also made significant contributions to adjoint methods for chaotic dynamical systems, including the development of stabilized approaches for sensitivity analysis and optimization of turbulent and chaotic flows, enabling reliable gradient computations for problems that were previously intractable using conventional adjoint techniques. His work integrates applied mathematics, computational science, and aerospace engineering to develop accurate, scalable, and industrially relevant algorithms for next-generation simulation and design.
Prof. Nadarajah serves as Associate Editor of the Journal of Optimization Theory and Applications and is an AIAA Associate Fellow. He has collaborated extensively with industry partners including Airbus, Bombardier, Pratt & Whitney Canada, CAE, and ANSYS, helping transition advanced computational methods into industrial aerospace applications. He has supervised numerous graduate students and postdoctoral researchers who now hold leadership positions in academia and industry worldwide, and regularly organizes major international conferences and workshops in computational fluid dynamics, high-order methods, and scientific computing.
Sensitivity analysis is of significant importance for optimization, uncertainty quantification, mesh adaptation, and flow control. At the same time, the predictive capability of these technologies depends critically on the robustness and accuracy of the underlying numerical discretization. Entropy-stable schemes have emerged as a powerful framework for high-fidelity turbulent flow simulation because they discretely satisfy the entropy inequality and provide nonlinear stability while preserving the fundamental mathematical structure of the governing equations. Beyond improving numerical robustness, entropy-stable methods offer a promising avenue for faithfully capturing the long-time dynamics of chaotic turbulent flows, including the growth of perturbations and the evolution of Lyapunov exponents that govern flow predictability.
While the conventional method of linearization and sensitivity analysis is well established for non-chaotic flows, it yields unbounded sensitivities for chaotic flows due to the presence of positive Lyapunov exponents [1]. Since high-fidelity scale-resolving simulations of turbulent flows are inherently chaotic, obtaining accurate sensitivities for such systems remains a fundamental challenge in computational fluid dynamics. Over the past decade, several shadowing-based methods have been developed to address this issue, including least-squares shadowing [2], non-intrusive least-squares shadowing [3], and stabilized march [4-5]. Among these approaches, the stabilized march method computes the shadow trajectory by imposing suitable boundary conditions on the adjoint differential equation and has been shown to converge to the correct long-time sensitivity as the integration time increases [4-5].
This keynote will present recent developments in adjoint-based sensitivity analysis for chaotic turbulent flows and examine their connection to stability and predictability theory. Applications of the stabilized march framework to three-dimensional turbulent flow simulations will be discussed, demonstrating accurate sensitivity predictions through comparisons with finite-difference calculations. The talk will also examine the role of entropy-stable high-order discretizations [6] in the computation of sensitivities and long-time statistics. Particular emphasis will be placed on the convergence of Lyapunov exponents, their interpretation as measures of predictability, and the influence of numerical discretization on the underlying chaotic dynamics. The impact of entropy-stable schemes, which are known to lack local energy stability [7], on sensitivity propagation and Lyapunov spectra will be explored. Together, these developments establish a unified perspective on sensitivity, stability, and chaos, providing new foundations for predictive simulation, optimization, and uncertainty quantification in turbulent-flow applications.
REFERENCES
[1] Lea, Daniel J. and Allen, Myles R. and Haine, Thomas W. N., Sensitivity analysis of the climate of a chaotic system, Tellus A: Dynamic Meteorology and Oceanography, 2000.
[2] Wang, Qiqi and Hu, Rui and Blonigan, Patrick, Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations, Journal of Computational Physics, Vol. 267, pp. 210-224, 2014.
[3] Ni, Angxiu and Wang, Qiqi, Sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Shadowing (NILSS), Journal of Computational Physics, Vol. 347, pp. 56-77, 2017.
[4] Thakur, Pranshul and Nadarajah, Siva, A stabilized march approach to adjoint-based sensitivity analysis of chaotic flows, arXiv preprint arXiv:2505.00838, 2025.
[5] Thakur, Pranshul and Nadarajah, Siva, Adjoint of Least Squares Shadowing: Existence, Uniqueness and Coarse Domain Discretization, Journal of Scientific Computing, 105(2), 47.
[6] Cicchino, Alexander and Nadarajah, Siva, Discretely nonlinearly stable weight-adjusted flux reconstruction high-order method for compressible flows on curvilinear grids, Journal of Computational Physics, Vol. 521, pp. 113532, 2025.
[7] Gassner, Gregor J and Svard, Magnus and Hindenlang, Florian J, Stability issues of entropy-stable and/or split-form high-order schemes: analysis of linear stability, Journal of Scientific Computing, Springer, Vol. 90, pp. 79, 2022.