Ritesh Kumar Dubey

Flux limited schemes: Their classification and accuracy based on total variation stability regions

A classification in terms of accuracy of flux limited high resolution schemes in steep gradient region is done which is based on two different total variation (TV) stability regions. The dependence of the TV stability regions on the smoothness parameter is shown. This dependence relation relates and pave a way to define a common unified TV stability region for both class of schemes. New flux limiters, satisfying the unified TV stability region are also proposed which are robust and work efficiently for both backward (left) and forward (right) moving solution profiles. Main significant feature of this classification is that it can be used to construct hybrid schemes and improve the accuracy of all existing flux limiters based schemes. Numerical results on linear test problems are given to support the theoretical discussion.

Low dissipative entropy stable schemes using third order WENO and TVD reconstructions

A low dissipative framework is given to construct high order entropy stable flux by addition of suitable numerical diffusion operator into entropy conservative flux. The framework is robust in the sense that it allows the use of high order reconstructions which satisfy the sign property only across the discontinuities. The third order weighted essentially non-oscillatory (WENO) interpolations and high order total variation diminishing (TVD) reconstructions are shown to satisfy the sign property across discontinuities. Third order accurate entropy stable schemes are constructed by using third order WENO and high order TVD reconstructions procedures in the diffusion operator. These schemes are efficient and less diffusive since the diffusion is actuated only in the sign stability region of the used reconstruction which includes discontinuities. Numerical results with constructed schemes for various test problems are given which show the third order accuracy and less dissipative nature of the schemes.

Suitable diffusion for constructing non-oscillatory entropy stable schemes

Abstract In this work, amount of suitable diffusion in entropy stable fluxes is explicitly characterized to construct non-oscillatory schemes in total variation diminishing (TVD) sense. Further, high resolution entropy stable TVD fluxes are constructed and a generic TVD-entropy stable region is given for the flux limiter functions. The non-oscillatory TVD property of proposed fluxes does not depend on the choice of entropy functions and different choices for diffusion matrices are proposed for these fluxes. These fluxes are extendable to the system of higher dimension and resulting entropy stable schemes are used to numerically compute the solution for Burgers and shallow water equations in 1D and 2D case. It is also shown numerically that, the use of proposed diffusion matrices in TECNO schemes can significantly suppress the oscillations exhibited by them while applied with other diffusion matrices. Numerical results show that the resulting schemes capture steady shock exactly and produce non-oscillatory solution profile with high resolution.

A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters

ABSTRACT In the present paper, a parameter-uniform numerical method is constructed and analysed for solving one-dimensional singularly perturbed parabolic problems with two small parameters. The solution of this class of problems may exhibit exponential (or parabolic) boundary layers at both the left and right part of the lateral surface of the domain. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution, we consider the implicit Euler method for time stepping on a uniform mesh and a special hybrid monotone difference operator for spatial discretization on a specially designed piecewise uniform Shishkin mesh. The resulting scheme is shown to be first-order convergent in temporal direction and almost second-order convergent in spatial direction. We then improve the order of convergence in time by means of the Richardson extrapolation technique used in temporal variable only. The resulting scheme is proved to be uniformly convergent of order two in both the spatial and temporal variables. Numerical experiments support the theoretically proved higher order of convergence and show that the present scheme gives better accuracy and convergence compared of other existing methods in the literature.

Stabilization and Best Actuator Location for the Navier--Stokes Equations

We study the numerical approximation of the boundary stabilization of the Navier--Stokes equations with mixed Dirichlet/Neumann boundary conditions, around an unstable stationary solution in a two dimensional domain. We first derive a semidiscrete controlled system, coming from a finite element approximation of the Navier--Stokes equations, which is new in the literature. We propose a new strategy for finding a boundary feedback control law able to stabilize the nonlinear semidiscrete controlled system in the presence of boundary disturbances. We determine the best control location. Next, we study the degree of stabilizability of the different real generalized eigenspaces of the controlled system. Based on that analysis, we determine an invariant subspace

A composite semi-conservative scheme for hyperbolic conservation laws

\mathbb{Z}_u

Local maximum principle satisfying high‐order non‐oscillatory schemes

and the projection of the controlled system onto

An Analysis on Induced Numerical Oscillations by Lax-Friedrichs Scheme

\mathbb{Z}_u

Accuracy Preserving ENO and WENO Schemes using Novel Smoothness Measurement.

. The projected system is used to determine feedback control laws. Our numerical results show that this control strategy is quite efficient when applied to the Navier--Stokes s...

An Investigation on Three Point Explicit Schemes and Induced Numerical Oscillations

In this work a first order accurate semi-conservative composite scheme is presented for hyperbolic conservation laws. The idea is to consider the non-conservative form of conservation law and utilize the explicit wave propagation direction to construct semi-conservative upwind scheme. This method captures the shock waves exactly with less numerical dissipation but generates unphysical rarefaction shocks in case of expansion waves with sonic points. It shows less dissipative nature of constructed scheme. In order to overcome it, we use the strategy of composite schemes. A very simple criteria based on wave speed direction is given to decide the iterations. The proposed method is applied to a variety of test problems and numerical results show accurate shock capturing and higher resolution for rarefaction fan.