Neeraj Sarna

Max Planck Institute for Dynamics of Complex Technical Systems | MPI · Department of Computational Methods and Systems Theory

We propose a data-driven model order reduction (MOR) technique for parametrized partial differential equations that exhibit parameter-dependent jump-discontinuities. Such problems have poor-approximability in a linear space and therefore, are challenging for standard MOR techniques. We build upon the methodology of approximating the map between the...

Given a set of solution snapshots of a hyperbolic PDE, we are interested in learning a reduced order model (ROM). To this end, we propose a novel decompose then learn approach. We decompose the solution by expressing it as a composition of a transformed solution and a de-transformer. Our idea is to learn a ROM for both these objects, which, unlike...

We propose a data-driven model order reduction (MOR) technique for parametrized partial differential equations that exhibit parameter-dependent jump-discontinuities. Such problems have poor-approximability in a linear space and therefore, are challenging for standard MOR techniques. We build upon the methodology of approximating the map between the...

We consider the method-of-moments approach to solve the Boltzmann equation of rarefied gas dynamics, which results in the following moment-closure problem. Given a set of moments, find the underlying probability density function. The moment-closure problem has infinitely many solutions and requires an additional optimality criterion to single-out a...

We are interested in solving the Boltzmann equation of chemically reacting rarefied gas flows using the Grad’s-14 moment method. We first propose a novel mathematical model that describes the collision dynamics of chemically reacting hard spheres. Using the collision model, we present an algorithm to compute the moments of the Boltzmann collision o...

We consider the method-of-moments approach to solve the Boltzmann equation of rarefied gas dynamics, which results in the following moment-closure problem. Given a set of moments, find the underlying probability density function. The moment-closure problem has infinitely many solutions and requires an additional optimality criterion to single-out a...

Snapshot matrices of hyperbolic equations have a slow singular value decay, resulting in inefficient reduced-order models. We develop on the idea of inducing a faster singular value decay by computing snapshots on a transformed spatial domain, or the so-called snapshot calibration/transformation. We are particularly interested in problems involving...

We are interested in solving the Boltzmann equation of chemically reacting rarefied gas flows using the Grad's-14 moment method. We first propose a novel mathematical model that describes the collision dynamics of chemically reacting hard spheres. Using the collision model, we present an algorithm to compute the moments of the Boltzmann collision o...

In (Commun Pure Appl Math 2(4):331-407, 1949), Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed boundary conditions lead to instabilities in Grad's Hermite expansion, which could result in non-converging solut...

A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The number of particles in a finite volume is bounded in time by the number of particles initially occupying the volume augmented by the total number of particles that entered the domain over time. In this paper, we present boundary conditions (BCs...

We propose an efficient residual minimization technique for the nonlinear model-order reduction of parameterized hyperbolic partial differential equations. Our nonlinear approximation space is a span of snapshots evaluated on a shifted spatial domain, and we compute our reduced approximation via residual minimization. To speed-up the residual minim...

Previous works have developed boundary conditions that lead to the $L^2$-boundedness of solutions to the linearised moment equations. Here we present a spatial discretization that preserves the $L^2$-stability by recovering integration-by-parts over the discretized domain and by imposing boundary conditions using a simultaneous-approximation-term (...

We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary condit...

Any numerical method fails to provide us with acceptable results if not equipped with appropriate boundary conditions. Catering to more realistic applications, in the present article we have extended the work done on the one plus one dimensional Boltzmann equation to the Boltzmann equation involving multi-dimensions in physical and velocity space....

An hierarchical simulation approach for Boltzmann's equation should provide single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics, but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. We use Herm...

For any study of microflows it is crucial to understand the collision dynamics of the molecules involved. In the present work we will discuss the collision dynamics of chemically reacting hard spheres(CRHS). The inability of the classical smooth inelastic hard spheres, which have been extensively used in the past to study granular gases, to describ...

The derivation of non-linear Grad's 2×26-moment (2×G26) equations for a binary gas-mixture of monatomic-inert-ideal hard sphere gases is sketched, although—for conciseness—only the linear 2×G26 equations are illustrated and analysed. The linear stability analysis is performed on 2×G26 equations by studying the dispersion relation and by considering...