Rakesh kumar

Rakesh kumar
TIFR CAM Bangalore · Mathematics

Ph. D

About

21
Publications
2,166
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90
Citations
Introduction
Rakesh kumar currently works at the Mathematics, TIFR CAM Bangalore. Rakesh does research in Applied Mathematics. Their most recent publication is 'Simple smoothness indicator and multi-level adaptive order WENO scheme for hyperbolic conservation laws'.

Publications

Publications (21)
Article
The Weighted Essentially Non-Oscillatory (WENO) scheme is an accurate and robust reconstruction procedure to simulate compressible flows, especially in the presence of discontinuities in the solution. The recently introduced multi-level WENO schemes like WENO-ZQ (Zhu \& Qiu, 2016) \cite{zhu-qiu_16a}, WENO-AO (Balsara et al.2016) \cite{bal-etal_16a}...
Article
In the present work, we construct a new, improved version of fifth-order finite-difference Weighted Essentially Non-Oscillatory (WENO) scheme with less dissipation to approximate solutions for one- and two- dimensional hyperbolic conservation laws and associated problems. The higher-order information presented for the classical WENO scheme is appli...
Article
TheWeighted Essentially Non-Oscillatory (WENO) reconstruction provides higher order accurate solutions to hyperbolic conservation laws for convex flux. But it fails to capture composite structure in the case of non-convex flux and converges to the wrong solution [Qiu & Shu SIAM J. Sci. Comput., 31 (2008), 584-607]. In this article, we have develope...
Conference Paper
Full-text available
In the present work, we have proposed a high order hybrid FDM- WENO method for the solution of convection-diffusion problems. In hybrid FDM-WENO method, a fifth order finite difference central flux is used to compute the convective flux in smooth regions whereas in a region where the solution has sharp variations or discontinuities, Weighted Essent...
Article
Full-text available
We propose a constraint preserving discontinuous Galerkin method for ideal compressible MHD in two dimensions and using Cartesian grids, which automatically maintains the global divergence-free property. The approximation of the magnetic field is achieved using Raviart–Thomas polynomials and the DG scheme is based on evolving certain moments of the...
Preprint
We propose a constraint preserving discontinuous Galerkin method for ideal compressible MHD in two dimensions and using Cartesian grids, which automatically maintains the global divergence-free property. The approximation of the magnetic field is achieved using Raviart-Thomas polynomials and the DG scheme is based on evolving certain moments of the...
Article
In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on kin...
Article
In this article an efficient sixth-order finite difference weighted essentially non-oscillatory scheme is developed to solve nonlinear degenerate parabolic equations. A new type of nonlinear weights are constructed with an introduction of a global smoothness indica- tor by a linear combination of local derivatives information involved in the smalle...
Article
Full-text available
We develop a positivity-preserving finite difference WENO scheme for the Ten-Moment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability...
Preprint
Full-text available
We develop a positivity-preserving finite difference WENO scheme for the Ten-Moment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability...
Preprint
We develop a positivity-preserving finite difference WENO scheme for the Ten-Moment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability...
Article
In this article, new efficient seventh order adaptive WENO schemes are proposed for hyperbolic conservation laws. The accuracy of proposed schemes are comparable with existing versions of seventh order adaptive WENO scheme known as WENO-AO(7,3) and WENO-AO(7,5,3) [Balsara, Garain, and Shu, J. Comput. Phys., 326 (2016), pp. 780–804]. The accuracy of...
Preprint
In the present work, we propose two new variants of fifth order finite difference WENO schemes of adaptive order. We compare our proposed schemes with other variants of WENO schemes with special emphasize on WENO-AO(5,3) scheme [Balsara, Garain, and Shu, {\it J. Comput. Phys.}, 326 (2016), pp 780-804]. The first algorithm (WENO-AON(5,3)), involves...
Article
In the present work, we propose two new variants of fifth order finite difference WENO schemes of adaptive order. We compare our proposed schemes with other variants of WENO schemes with special emphasize on WENO-AO(5,3) scheme (Balsara et al., 2016) [3]. The first algorithm (WENO-AON(5,3)), involves the construction of a new simple smoothness indi...
Article
In general, B-spline quasi-interpolation (BSQI)-based numerical schemes for hyperbolic conservation laws are unstable in nature. In the present work, we have developed the stable modified version of the cubic B-spline quasi-interpolation (CBSQI) numerical scheme for the hyperbolic conservation laws in one space dimension. In order to stabilize the...
Article
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-oscillator...
Article
Abstract In this article, we intend to use quadratic and cubic B-spline quasi-interpolants to develop higher order numerical methods for some Sobolev type equations in one space dimension. Our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence. We also discuss another approach to the...
Poster
Full-text available
In the present work, we propose B-Spline Quasi-Interpolation (BSQI) based numerical schemes for the convection-diffusion equation. Approximation of a function and its derivatives by corresponding BSQI is already established. The main idea of the numerical scheme is to replace the first and the second derivatives of the solution with the derivatives...
Poster
Full-text available
The present work analyzes the B-Spline Quasi-Interpolation (BSQI) based explicit numerical schemes for hyperbolic conservation laws. Approximation of a function and its derivatives by corresponding BSQI is already established (see Sablonnì ere [4]). The BSQI has better approximation of the derivative in comparison with finite difference approximati...
Poster
Full-text available
In this poster, we present Quadratic B-Spline Quasi-Interpolation (QBSQI) based second order accurate numerical scheme in space for some non-linear wave equations. We approximate the space derivative of dependent variable by the derivative of QBSQI and for temporal derivative Euler's forward formula is considered. The L 2 stability of the proposed...

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Project (1)
Project
The project goal is to construct efficient and high-resolution numerical schemes for conservation laws and its related applications like nonlinear degenerate parabolic PDE's, Hamilton-Jacobi equations, MHD equations, convection-diffusion equations, etc.