Center for Applicable Mathematics

Bangalore, India

Center for Applicable Mathematics

Bangalore, India
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Meena A.K.,Indian Institute of Technology Delhi | Kumar H.,Indian Institute of Technology Delhi | Chandrashekar P.,Center for Applicable Mathematics
Journal of Computational Physics | Year: 2017

Euler equations for compressible flows treats pressure as a scalar quantity. However, for several applications this description of pressure is not suitable. Many extended model based on the higher moments of Boltzmann equations are considered to overcome this issue. One such model is Ten-Moment Gaussian closure equations, which treats pressure as symmetric tensor. In this work, we develop a higher-order, positivity preserving Discontinuous Galerkin (DG) scheme for Ten-Moment Gaussian closure equations. The key challenge is to preserve positivity of density and symmetric pressure tensor. This is achieved by constructing a positivity limiter. In addition to preserve positivity, the scheme also ensures the accuracy of the approximation for smooth solutions. The theoretical results are then verified using several numerical experiments. © 2017 Elsevier Inc.

Kumar R.,Center for Applicable Mathematics
BIT Numerical Mathematics | Year: 2017

In this article, we have proposed a septic B-spline quasi-interpolation (SeBSQI) based numerical scheme for the modified Burgers’ equation. The SeBSQI scheme maintains eighth order accuracy for the smooth solution, but fails to maintain a non-oscillatory profile when the solution has discontinuities or sharp variations. To ensure the non-oscillatory profile of the solution, we have proposed an adaptive SeBSQI (ASeBSQI) scheme for the modified Burgers’ equation. The ASeBSQI scheme maintains higher order accuracy in the smooth regions using SeBSQI approximation and in regions with discontinuities or sharp variations, 5th order weighted essentially non-oscillatory (WENO) reconstruction is used to preserve a non-oscillatory profile. To identify discontinuous or sharp variation regions, a weak local truncation error based smooth indicator is proposed for the modified Burgers’ equation. For the temporal derivative, we have considered the Runge–Kutta method of order four. We have shown numerically that the ASeBSQI scheme preserves the convergence rate of the SeBSQI and it converges to the exact solution with convergence rate eight. We have performed numerical experiments to validate the proposed scheme. The numerical experiments demonstrate an improvement in accuracy and efficiency of the proposed schemes over the WENO5 and septic B-spline collocation schemes. The ASeBSQI scheme is also tested for one-dimensional Euler equations. © 2017 Springer Science+Business Media B.V.

Apte A.,Center for Applicable Mathematics | Jones C.K.R.T.,University of North Carolina at Chapel Hill
Nonlinear Processes in Geophysics | Year: 2013

The focus of this paper is on how two main manifestations of nonlinearity in low-dimensional systems-shear around a center fixed point (nonlinear center) and the differential divergence of trajectories passing by a saddle (nonlinear saddle)-strongly affect data assimilation. The impact is felt through their leading to non-Gaussian distribution functions. The major factors that control the strength of these effects is time between observations, and covariance of the prior relative to covariance of the observational noise. Both these factors-less frequent observations and larger prior covariance-allow the nonlinearity to take hold. To expose these nonlinear effects, we use the comparison between exact posterior distributions conditioned on observations and the ensemble Kalman filter (EnKF) approximation of these posteriors. We discuss the serious limitations of the EnKF in handling these effects. © 2013 Author(s).

Chandrashekar P.,Center for Applicable Mathematics
Journal of Computational Physics | Year: 2013

Kinetic schemes for compressible flow of gases are constructed by exploiting the connection between Boltzmann equation and the Navier-Stokes equations. This connection allows us to construct a flux splitting for the Navier-Stokes equations based on the direction of molecular motion from which a numerical flux can be obtained. The naive use of such a numerical flux function in a discontinuous Galerkin (DG) discretization leads to an unstable scheme in the viscous dominated case. Stable schemes are constructed by adding additional terms either in a symmetric or non-symmetric manner which are motivated by the DG schemes for elliptic equations. The novelty of the present scheme is the use of kinetic fluxes to construct the stabilization terms. In the symmetric case, interior penalty terms have to be added for stability and the resulting schemes give optimal convergence rates in numerical experiments. The non-symmetric schemes lead to a cell energy/entropy inequality but exhibit sub-optimal convergence rates. These properties are studied by applying the schemes to a scalar convection-diffusion equation and the 1-D compressible Navier-Stokes equations. In the case of Navier-Stokes equations, entropy variables are used to construct stable schemes. © 2012 Elsevier Inc.

Chandrashekar P.,Center for Applicable Mathematics
Communications in Computational Physics | Year: 2013

Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form {equation presented} are any consistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes. © 2013 Global-Science Press.

Maity D.,Center for Applicable Mathematics
ESAIM - Control, Optimisation and Calculus of Variations | Year: 2015

In this article, we study the null controllability of linearized compressible Navier-Stokes system in one and two dimension. We first study the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around (ρ, 0, θ), with ρ > 0, θ > 0 is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove that the the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state (ρ, υ, θ), with ρ > 0, υ > 0, θ > 0 is not null controllable by localized interior control or by boundary control for small time T. Next we consider two-dimensional compressible Navier-Stokes system for barotropic fluid linearized around a constant steady state (ρ, 0). We prove that this system is also not null controllable by localized interior control. © EDP Sciences, SMAI 2015.

Biswas A.,Center for Applicable Mathematics
Systems and Control Letters | Year: 2011

Risk sensitive control problem under near monotonicity condition is considered. We prove existence of a solution to the corresponding HamiltonJacobiBellman equation by an eigenvalue approach. Existence of an optimal control has also been proved. © 2010 Elsevier B.V.

Apte A.,Center for Applicable Mathematics
Physica D: Nonlinear Phenomena | Year: 2011

Renormalization group (RG) methods provide a unifying framework for understanding critical behaviour, such as transition to chaos in area-preserving maps and other dynamical systems, which have associated with them universal scaling exponents. Recently, de la Llave et al. (2007) [10] have formulated the Principle of Approximate Combination of Scaling Exponents (PACSE for short), which relates exponents for different criticalities via their combinatorial properties. The main objective of this paper is to show that certain integrable fixed points of RG operators for area-preserving maps do indeed follow the PACSE. © 2010 Elsevier B.V. All rights reserved.

Duraisamy K.,Stanford University | Chandrashekar P.,Center for Applicable Mathematics
Computers and Fluids | Year: 2012

We propose a framework based on the use of adjoint equations to formulate an adaptive sampling strategy for uncertainty quantification for problems governed by algebraic or differential equations involving random parameters. The approach is non-intrusive and makes use of discrete sampling based on collocation on simplex elements in stochastic space. Adjoint or dual equations are introduced to estimate errors in statistical moments of random functionals resulting from the inexact reconstruction of the solution within the simplex elements. The approach is demonstrated to be accurate in estimating errors in statistical moments of interest and shown to exhibit super-convergence, in accordance with the underlying theoretical rates. Goal-oriented error indicators are then built using the adjoint solution and exploited to identify regions for adaptive sampling. The error-estimation and adaptive refinement strategy is applied to a range of problems including those governed by algebraic equations as well as scalar and systems of ordinary and partial differential equations. The strategy holds promise as a reliable method to set and achieve error tolerances for efficient aleatory uncertainty quantification in complex problems. Furthermore, the procedure can be combined with numerical error estimates in physical space so as to effectively manage a computational budget to achieve the best possible overall accuracy in the results. © 2012 Elsevier Ltd.

Chandrashekar P.,Center for Applicable Mathematics | Garg A.,Center for Applicable Mathematics
Computers and Mathematics with Applications | Year: 2013

Vertex-centroid schemes are cell-centered finite volume schemes for conservation laws which make use of both centroid and vertex values to construct high-resolution schemes. The vertex values must be obtained through a consistent averaging (interpolation) procedure while the centroid values are updated by the finite volume scheme. A modified interpolation scheme is proposed which is better than existing schemes in giving positive weights in the interpolation formula. A simplified reconstruction scheme is also proposed which is also more efficient and leads to more robust schemes for discontinuous problems. For scalar conservation laws, we develop limited versions of the schemes which are stable in maximum norm by constructing suitable limiters. The schemes are applied to compressible flows governed by the Euler equations of inviscid gas dynamics. © 2012 Elsevier Ltd. All rights reserved.

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