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Title:  Accurate Computational Algorithms For Hyperbolic Conservation Laws 
Authors:  Jaisankar, S 
Advisors:  Rao, S V Raghurama 
Keywords:  Gas Dynamics Magnetohydrodynamics Conservation Laws Algorithms Numerical Analysis Diffusion (Mathematical Physics) Hyperbolic Equations (Mathematical Analysis) Diffusion Regulator Model Hyperbolic Partial Differential Equations Compressible Flows  Numerical Methods Hyperbolic Consevation Laws Diffusion Regulated Schemes UpwindBiased Scheme Rankine Hugoniot Solver Gridfree Central Solver 
Submitted Date:  Jul2008 
Series/Report no.:  G22941 
Abstract:  The numerics of hyperbolic conservation laws, e.g., the Euler equations of gas dynamics, shallow water equations and MHD equations, is nontrivial due to the convective terms being highly nonlinear and equations being coupled. Many numerical methods have been developed to solve these equations, out of which central schemes and upwind schemes (such as Flux Vector Splitting methods, Riemann solvers, Kinetic Theory based Schemes, Relaxation Schemes etc.) are well known. The majority of the above mentioned schemes give rise to very dissipative solutions. In this thesis, we propose novel low dissipative numerical algorithms for some hyperbolic conservation laws representing fluid flows. Four different and independent numerical methods which give low diffusive solutions are developed and demonstrated.
The first idea is to regulate the numerical diffusion in the existing dissipative schemes so that the smearing of solution is reduced. A diffusion regulator model is developed and used along with the existing methods, resulting in crisper shock solutions at almost no added computational cost. The diffusion regulator is a function of jump in Mach number across the interface of the finite volume and the average Mach number across the surface. The introduction of the diffusion regulator makes the diffusive parent schemes to be very accurate and the steady contact discontinuities are captured exactly. The model is demonstrated in improving the diffusive Local LaxFriedrichs (LLF) (or Rusanov) method and a Kinetic Scheme. Even when employed together with accurate methods of Roe and Osher, improvement in solutions is demonstrated for multidimensional problems.
The second method, a Central UpwindBiased Scheme (CUBS), attempts to reorganize a central scheme such that information from irrelevant directions is largely reduced and the upwind biased information is retained. The diffusion coefficient follows a new format unlike the use of maximum characteristic speed in the Local LaxFriedrichs method and the scheme results in improved solutions of the flow features. The gridaligned steady contacts are captured exactly with the reorganized format of diffusion coefficient. The stability and positivity of the scheme are discussed and the procedure is demonstrated for its ability to capture all the features of solution for different flow problems.
Another method proposed in this thesis, a Central RankineHugoniot Solver, attempts to integrate more physics into the discretization procedure by enforcing a simplified RankineHugoniot condition which describes the jumps and hence resolves steady discontinuities very accurately. Three different variants of the scheme, termed as the Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), based on a single wave (MOVERS1), multiple waves (MOVERSn) and limiter based diffusion (MOVERSL) are presented. The scheme is demonstrated for scalar Burgers equation and systems of conservation laws like Euler equations, ideal Magnetohydrodynamics equations and shallow water equations. The new scheme uniformly improves the solutions of the Local LaxFriedrichs scheme on which it is based and captures steady discontinuities either exactly or very accurately.
A GridFree Central Solver, which does not require a grid structure but operates on any random distribution of points, is presented. The gridfree scheme is generic in discretization of spatial derivatives with the location of the midpoint between a point and its neighbor being used to define a relevant coefficient of numerical dissipation. A new central scheme based on convectivepressure splitting to solve for midpoint flux is proposed and many test problems are solved effectively. The RankineHugoniot Solver, which is developed in this thesis, is also implemented in the gridfree framework and its utility is demonstrated.
The numerical methods presented are solved in a finite volume framework, except for the GridFree Central Solver which is a generalized finite difference method. The algorithms developed are tested on problems represented by different systems of equations and for a wide variety of flow features. The methods presented in this thesis do not need any eigenstructure and complicated flux splittings, but can still capture discontinuities very accurately (sometimes exactly, when aligned with the grid lines), yielding low dissipative solutions.
The thesis ends with a highlight on the importance of developing genuinely multidimensional schemes to obtain accurate solutions for multidimensional flows. The requirement of simpler discretization framework for such schemes is emphasized in order to match the efficacy of the popular dimensional splitting schemes. 
URI:  http://hdl.handle.net/2005/905 
Appears in Collections:  Aerospace Engineering (aero)

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