# Contributed Papers

Sangita Yadav, BITS Pilani, SUPERCONVERGENCE OF A CLASS OF EXPANDED DISCONTINUOUS GALERKIN METHODS FOR FULLY NONLINEAR ELLIPTIC PROBLEMS IN DIVERGENCE FORM

Abstract: For fully nonlinear elliptic boundary value problems in divergence form, improved error estimates are derived in the frame work of a class of expanded discontinuous Galerkin methods. It is shown that the error estimate for the discrete flux in $L^2$ -norm is of order $k + 1$, when piecewise polynomials of degree $k ≥ 1$ are used to approximate both potential as well as flux variables. Then, solving a discrete linear elliptic problem in each element locally, a suitable post-processing of the discrete potential is proposed and it is proved that the resulting post-processed potential converges with order of convergence $k + 2$ in $L^2$ -norm. By choosing stabilizing parameters appropriately, similar results are derived for the expanded HDG methods for nonlinear elliptic problems.

Samala Rathan, Visvesvaraya National Institute of Technology, Nagpur, Investigation of weighted essentially non-oscillatory scheme at critical points and near discontinuities

Abstract: We analyse the fifth order weighted essentially non-oscillatory (WENO-5) scheme proposed by Jiang and Shu [1] near discontinuities. At a transition point, from smooth region to a discontinuity or a discontinuity to smooth region, the accuracy of WENO-5 scheme reduces to third order. The weights which are defined in the classical sense lose accuracy at critical points (f'_{i} = 0) also. The order of accuracy of the scheme can be increased by combining the high order smoothness indicators with fourth order fluxes which was proposed by Shen and Zha [4] near discontinuities. It is found that the more accurate results with newly proposed modified WENO-neta_tau(8) and modified WENO-Z+ schemes. The numerical examples are given to support and compared with the WENO-Z [2], WENO-Z+ [5], WENO-neta_tau(5) [3] schemes. *The references are provided in the abstract of the paper which was provided earlier.

SHUVAM SEN, Tezpur University, On the development of a nonprimitive Navier-Stokes formulation subject to rigorous implementation of a new vorticity integral condition

Abstract: A new integral vorticity boundary condition has been developed and implemented to compute solution of non-primitive Navier-Stokes equations. Global integral vorticity condition which is of primitive character can be considered to be of entirely different kind compared to other vorticity conditions that are used for computation in literature. The main purpose of this work is to design an algorithm which is easy to implement and explicit. The competency of the proposed boundary methodology vis-a vis other popular vorticity boundary conditions has been amply appraised by its use in a model problem that embodies the essential features of the incompressibility and viscosity. Finally, it has been applied to three benchmark problems governed by the incompressible Navier-Stokes equations viz., lid driven cavity, backward facing step and flow past a circular cylinder.

Biswarup Biswas, SRM University, An approach to construct high order entropy consistent schemes for system of conservation laws

Abstract: An approach is given to construct entropy consistent high order (ECHO) schemes by using any high order reconstruction. The main strength of the proposed approach is that it allows the use of non sign-preserving high order reconstruction procedure for defining higher order diffusion operator. In order to show the robustness of the proposed approach, a fifth order entropy consistent scheme (termed as ECHO-WENO5 ) constructed using a non sign preserving WENO interpolation.

Asha Kumari Meena, IIT Delhi, Positive Second-order MUSCL Scheme for Ten-moment Gaussian Closure Model

Abstract: Ten-moment Gaussian closure model is obtained by velocity moments of Boltzmann equation. Local thermodynamics equilibrium is not assumed in this model unlike Euler equations of compressible flow. The system of Ten-moment equations is hyperbolic provided density and pressure tensor are positive. In this work, we obtain a second-order MUSCL scheme for two-dimension Ten-moment model, which is also positivity preserving. Here second-order accuracy is achieved by linear reconstruction process. For positivity of the scheme, slop limiters are introduced in the reconstruction process. We introduce reconstruction process in primitive variables with general slop and conservative slop both. Resultant numerical schemes were tested on several practical examples to check robustness of the schemes over standard second-order schemes.

P. Dhanumjaya, BITS-Pilani Goa, Orthogonal Cubic Spline Collocation Methods for the One-Dimensional Helmholtz equation with discontinuous coefficients

Abstract: In this paper, we use orthogonal spline collocation methods for one-dimensional and two-dimensional Helmholtz equation with discontinuous coefficients. We use cubic monomial and piecewise Hermite cubic basis functions to approximate the solution. Finally, we perform several numerical experiments with different wave numbers and using grid refinement analysis, we compute the order of convergence of the numerical scheme.

Saurav Samantaray, IISER Trivandrum, Stiffly Accurate Implicit-Explicit Runge-Kutta Schemes for Multiscale Problems

Abstract: Mathematical models of several physical problems admit multiple time and space scales, e.g. hyperbolic problems with relaxation, compressible and geophysical fluid flows etc. The mathematical difficulty is associated with the singular limits which arise as the governing equations change their nature. Numerical schemes suffer from severe pathologies, such as stiffness, reduction of order etc. when a singular limit is taken. We consider the recently introduced Implicit-Explicit Runge-Kutta (IMEX RK) schemes developed by Ascher et al. The right hand side of the ODE system is split into a non-stiff and stiff terms with the stiff terms involving a singular parameter, the non-stiff terms are usually discretised by a high order explicit RK method and the stiff terms by an implicit RK method. To simplify the implicit nature of the scheme and the solution of the resulting algebraic equations, only diagonally implicit RK schemes are usually studied. We review the A-stability and L-stability of some of the IMEX-RK schemes and present the application of IMEX RK schemes to stiff limits of hyperbolic systems. In particular, we focus on stiff wave equations, singularly perturbed advection-diffusion equations etc. Our focus will be on effectively splitting the equations into stiff and non-stiff parts which facilitates the use of IMEX schemes. We present the results of numerical case studies which confirm the efficacy of our flux splittings and accuracy and stability of the resulting IMEX RK schemes.

Arnab Jyoti Das Gupta, IISER Trivandrum, Analysis of an assymptotic preserving semi-implicit scheme for the wave equation system in the low Mach number limit

Abstract: Low-mach number limit is a singular limit for the purely hyperbolic compressible equations which change their type to mixed hyperbolic-elliptic equations. Due to this assymptotic convergence, numerical schemes designed only for compressible equations suffer from several pathologies: namely, reduction of order, lack of stability and inconsistency. In the present work we consider a framework for analysing the above three difficulties by taking the wave equation with advection as a prototype model. Guided by a systematic multiscale asymptotic expansion, we split the fluxes into so called stiff and non-stiff parts. The non-stiff part is so designed that its Jacobian matrix always has finite eigenvalues, facilitating the use of standard upwind discretizing procedures. On the other hand, the stiff part constitutes a wave equation system with very large wave speeds. We use an implicit-explicit (IMEX) Runge-Kutta scheme for discretization, therein the non-stiff part is treated explicitly and the stiff part implicitly.

Balaji Srinivasan, IIT Delhi, On the entropy and dissipation of artificial diffusion schemes

Abstract: We discuss certain paradoxical behavior of artificial diffusion schemes. We show that the Burgers sonic glitch can be seen as locally entropy violating behavior. This leads to the following paradox -- even entropy satisfying schemes such as Godunov can show locally entropy violating behavior. We show a simple, dissipation based generalization of the entropy idea and show that artificial diffusion schemes hides within themselves entropy violating behavior in their very structure.

S. Gowrisankar, IIT Guwahati, Uniformly Convergent Numerical Methods for Singularly Perturbed Burger’s Equation

Abstract: In this work, we consider singularly perturbed Burgers’ equation. The Burger’s equation is a fundamental partial differential equation from fluid mechanics. It also occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow. To solve the singularly perturbed Burgers equation, we use linearization process and obtain the Newton sequence. And finite difference scheme, which comprises of the backward difference for time derivative and the upwind difference for the spatial derivatives, is applied to the linear PDEs. The stability of the difference scheme is studied and ε-uniform error estimate of O(N-1+ ∆t) is obtained. Numerical experiments are carried out to validate the theoretical results.

Chinedu Nwaigwe, University of Warwick, Dynamic Approach for Coupling 2D/1D Shallow Water Equations using Finite Volume Methods

Siddesh Desai, IIT Guwahati, Shock Wave Boundary Layer Interaction with Real Gas Effects

Abstract: Challenges in the field of hypersonic aerodynamics are inevitably due to high temperature effects. Chemical and thermal non-equilibrium effects complicate the understanding of the flow field in any application in this flow regime. Due to this, most of the investigations for the topic of shock wave boundary layer interaction (SWBLI) deal with perfect gas assumption. Prediction of various parameters like upstream influence location, separation bubble size, peak pressure and peak heat transfer rate becomes difficult in the presence of real gas effects. In view of importance of thermal and chemical non-equilibrium effects and associated scarcity of data for laminar high enthalpy flow, a finite volume based unstructured flow solver has been developed to account for change in specific heat with temperature and chemical reactions. Present solver considers five species (N2,O2, NO, O and N) and prominent eleven reactions. In the absence of reaction, this solver can simulate the frozen flow condition. This solver is equipped with five inviscid flux computing schemes viz. AUSM, AUSM+, RUSNOV, Roe and Van Leer. Initially, studies are carried out to validate the results of the present solver for inviscid flow and then for the viscous flow test cases reported in the literature. This validated solver is then considered to study the non-equilibrium effect in laminar shock wave boundary layer interaction. Prediction of features of this interaction is found to be strongly dependent on non-equilibrium effects. All the inviscid flux schemes of the solver are considered for such predictions. Thus present studies highlight necessity of investigations of SWBLI in the presence of real gas effects. Besides this, it also assesses applicability of the chosen schemes for high enthalpy separated flow conditions.

Raushan Kumar, IIT Guwahati, A new kinetic theory based flux-limited numerical scheme for 1-d Euler equations of gas dynamics

Abstract: A new kinetic theory based scheme for the 1-D Euler equations of gas dynamics is formulated which uses the flux-limited approach of computational gas dynamics. The assumption of local equilibrium in a flow of dilute gas results in making the collision term in the Boltzmann equation vanish as the depleting and replenishing collisions balance locally. Taking moment of this collisionless Boltzmann equation with Maxwellian distribution gives Euler equations of gas dynamics. This connection between Boltzmann equation and Euler equations has been utilized in developing many kinetic schemes for the solution of Euler equations. Kinetic numerical method (KNM) and kinetic flux vector splitting (KFVS) scheme are examples of some such schemes. Particularly in KFVS the Boltzmann equation is discretized using Courant-Issacsion-Rees (CIR) method and then moment of this equation is taken to get an upwind scheme for the Euler equations. This scheme is highly dissipative and smears the discontinuities like shocks and contacts badly. There are two obvious reasons for smearing. First is the truncation error (numerical viscosity) and the second is the use of wrong governing equation. The collisionless Boltzmann equation by itself does not possess any mechanism which can make the flow to come to local equilibrium once it started from local equilibrium. For this reason analytic solution (if possible) of a flow situation with the collisionless Boltzmann equation and the Euler equations will be different.

Maruthi N.H, IISc, A Hybrid Central Solver for Compressible Euler Equations

Abstract: In this work, a hybrid solver for compressible Euler equations is introduced. The MOVERS (Method of Optimal Viscosity for Enhanced Resolution of Shocks) scheme developed by Jaisankar and Raghurama Rao (J Computat Phys 2009;228:770-798) is an accurate scheme capable of capturing grid aligned steady shocks and steady contact discontinuities exactly. However, this being an accurate scheme has low numerical diffusion and suffers from mild sonic glitch and prone to shock instabilities, thereby requiring entropy fix. Although, sonic glitch and shock instabilities are avoided with the use of entropy fix, it will loose its ability to capture steady discontinuities exactly. MOVERS scheme is further improved by using pressure based dissipation scaled by maximum eigenvalue along with MOVERS dissipation. This new resulting scheme captures steady contact discontinuities exactly and avoids shock instabilities. The new method termed MOVERS-H (H stands for Hybrid) scheme is tested on various benchmark problems for accuracy, robustness and for its ability to capture flow features without instabilities.

Pritam Giri, IISc, Isotropic finite volume projection method for the incompressible Navier-Stokes equations

Abstract: We develop a conservative isotropic finite volume projection scheme for the simulation of viscous incompressible flows. To eliminate spurious oscillations that feature in a collocated primitive variable formulation, we adopt a fully staggered arrangement for the velocity and pressure fields for spatial discretization. We develop isotropic approximations of the relevant cell face line integrals along both intercell interfaces and the cell center locations to eliminate anisotropy from the leading order terms in the truncation error. Furthermore, general isotropic approximations for the nonlinear terms that allow for a directionally unbiased discretization of the convective terms in the momentum equation are derived. Results from comparative tests aimed at assessing the directional attributes of the isotropic method and contrasting them with the conventional formulation based on dimension-by-dimension extension of the one-dimensional sweeps are presented.

Sunil Kumar, National Institute of Technology Delhi, A new convergence proof of overlapping domain decomposition methods for singularly perturbed parabolic systems

Abstract: We develop an overlapping Schwarz domain decomposition method for numerically solving time dependent singularly perturbed parabolic reaction-diffusion systems. The discrete Schwarz method invokes two boundary layer subdomains and three interior subdomains with narrow subdomain overlaps of small widths. In the iterative steps of our method we employ the central difference scheme in spatial direction and the backward Euler scheme in time direction for the discretization of the continuous problem on each subdomain. Then we present a new technique for error analysis of the discrete Schwarz method which is based on defining some auxiliary problems that allows to prove the uniform convergence in two steps, splitting the discretization error and the iteration error. The numerical approximations obtained from the iterative method are shown to be uniformly convergent of order almost two in spatial direction and of order one in time direction. Numerical results are given in support of the theory.

Jayesh Badwaik, TIFR-CAM, ALE DG scheme for 1-D Euler equations

Abstract: We propose an explicit in time discontinuous Galerkin scheme on moving grids using the arbitrary Lagrangian-Eulerian approach for one dimensional Euler equations. The grid is moved with a velocity which is close to the local fluid velocity which considerably reduces the numerical dissipation in the Riemann solvers. Several test cases are provided to demonstrate the accuracy of the proposed scheme