Complex multiphysics systems—spanning electrical circuits, multibody dynamics, and computational fluid dynamics—are universally modeled using Differential-Algebraic Equations (DAEs). By combining differential equations for system evolution with algebraic constraints, DAEs capture the behavior of interacting physical components. However, for large-scale systems, traditional DAE formulations can obscure vital physical properties such as energy exchange and dissipation.
To shed light on this challenge, today we highlight the PhD project of M.Sc. Aashutosh Sharma, a doctoral researcher embedded within the DFG-funded Collaborative Research Centre (CRC) 1701 “Port-Hamiltonian Systems.” Aashutosh is jointly supervised by Andreas Bartel from the University of Wuppertal and Manuel Schaller from the Chemnitz University of Technology (Germany).
The Power of Port-Hamiltonian DAEs
When modeling coupled physical components, standard numerical methods can easily produce solutions that are mathematically accurate over short intervals yet exhibit qualitatively incorrect physical behavior over time.
Port-Hamiltonian DAEs (pH-DAEs) offer a powerful remedy. By representing subsystems through stored energy, dissipation, constraints, and energy-exchange ports, pH-DAEs provide a systematic framework for interconnection. This ensures that the physical structure of the continuous model is respected.
To maintain this fidelity numerically, researchers rely on structure-preserving time-integration methods (such as discrete-gradient methods and collocation schemes) that retain energy balances, passivity, and constraints at the discrete level. In his earlier work, Aashutosh investigated goal-oriented time adaptivity to construct time grids that achieve an optimal approximation of the underlying Hamiltonian [4].
Overcoming the Computational Bottleneck
While structure-preserving methods are mathematically robust, they introduce a significant computational hurdle: nonlinear algebraic systems that must be solved at every time step.
Typically handled by Newton-type methods, these solves require:
- The frequent construction or update of Jacobian matrices,
- The solution of large linear systems, and
- Additional globalization mechanisms (like line searches or trust-region strategies).
For large-scale simulations running over thousands of steps, the cost and variability of these nonlinear solves quickly become a major bottleneck.
A Generalized Scalar Auxiliary Variable (SAV) Approach
To bypass expensive Newton-type iterations while preserving physical structure, Aashutosh and his supervisors turned to an ingenious strategy: adapting the philosophy of the Scalar Auxiliary Variable (SAV) approach [5]—originally designed for gradient-flow problems—to nonlinear pH-DAEs [3].
This new framework offers distinct computational advantages:
- Predictable Workflows: It completely avoids nonlinear algebraic solvers at every time step, organizing computational work around a predictable sequence of linear-algebra operations.
- Hardware Compatibility: Fixed computational patterns and reusable matrix structures facilitate vectorization, cache-aware implementations, and pipelining.
- High-Performance Scalability: Predictable kernels and reusable operators are much easier to distribute across modern multi-core architectures and high-performance computing (HPC) compute nodes.
By bridging rigorous geometric physics with modern HPC architecture design, Aashutosh’s research paves the way for scalable, highly efficient simulations of complex industrial systems.
References & Further Reading
- P. Kunkel and V. Mehrmann, Differential-algebraic equations, European Mathematical Society, 2006.
- R. I. McLachlan, G. R. W. Quispel, and N. Robidoux, Geometric integration using discrete gradients, Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 1999.
- A. Sharma, A. Bartel, and M. Schaller, A Generalized Scalar Auxiliary Variable Method for Structure-Preserving and Efficient Integration of Nonlinear port-Hamiltonian DAEs, arXiv preprint arXiv:2609.05246, 2026.
- A. Sharma, A. Bartel, and M. Schaller, A Goal-Oriented Time Adaptivity for Linear Port-Hamiltonian Differential-Algebraic Equations of Index 1, arXiv preprint arXiv:2605.14082, 2026.
- J. Shen, J. Xu, and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, Journal of Computational Physics, 2018.
- A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Foundations and Trends in Systems and Control, 2014.
Are you working on numerical methods for industrial applications or port-Hamiltonian systems that you would like to feature on the ECMI blog? Let us know!