In this article, we propose a new grid-free and exact solution method for computing solutions associated with an hybrid traffic flow model based on the Lighthill- Whitham-Richards (LWR) partial differential equation. In this hybrid flow model, the vehicles satisfy the LWR equation whenever possible, and have a fixed acceleration otherwise. We first present a grid-free solution method for the LWR equation based on the minimization of component functions. We then show that this solution method can be extended to compute the solutions to the hybrid model by proper modification of the component functions, for any concave fundamental diagram. We derive these functions analytically for the specific case of a triangular fundamental diagram. We also show that the proposed computational method can handle fixed or moving bottlenecks.

Least Squares Kinetic Upwind Method (LSKUM), a grid free method based on kinetic schemes has been gaining popularity over the conventional CFD methods for computation of inviscid and viscous compressible flows past complex configurations. The main reason for the growth of popularity of this method is its ability to work on any point distribution. The grid free methods do not require the grid for flow simulation, which is an essential requirement for all other conventional CFD methods. However, they do require point distribution or a cloud of points. Point generation is relatively simple and less time consuming to generate as compared to grid generation. There are various methods for point generation like an advancing front method, a quadtree based point generation method, a structured grid generator, an unstructured grid generator or a combination of above, etc. One of the easiest ways of point generation around complex geometries is to overlap the simple point distributions generated around individual constituent parts of the complex geometry. The least squares grid free method has been successfully used to solve a large number of flow problems over the years. However, it has been observed that some problems are still encountered while using this method on point distributions around complex configurations. Close analysis of the problems have revealed that bad connectivity of the nodes is the cause and this leads to bad connectivity related code divergence. The least squares (LS) grid free method called LSKUM involves discretization of the spatial derivatives using the least squares approach. The formulae for the spatial derivatives are obtained by minimizing the sum of the squares of the error, leading to a system of linear algebraic equations whose solution gives us the formulae for the spatial derivatives. The least squares matrix A for 1-D and 2-D cases respectively is given by (Refer PDF File for equation) The 1-D LS formula for the spatial derivatives is always well behaved in the sense that ∑∆xi2 can never become zero. In case of 2-D problems can arise. It is observed that the elements of the Ls matrix A are functions of the coordinate differentials of the nodes in the connectivity. The bad connectivity of a node thus can have an adverse effect on the nature of the LS matrices. There are various types of bad connectivities for a node like insufficient number of nodes in the connectivity, highly anisotropic distribution of nodes in the connectivity stencil, the nodes falling nearly on a line (or a plane in 3-D), etc. In case of multidimensions, the case of all nodes in a line will make the matrix A singular thereby making its inversion impossible. Also, an anisotropic distribution of nodes in the connectivity can make the matrix A highly illconditioned thus leading to either loss in accuracy or code divergence. To overcome this problem, the approach followed so far is to modify the connectivity by including more neighbours in the connectivity of the node. In this thesis, we have followed a different approach of using weights to alter the nature of the LS matrix A. (Refer PDF File for equation) The weighted LS formulae for the spatial derivatives in 1-D and 2-D respectively are are all positive. So we ask a question : Can we reduce the multidimensional LS formula for the derivatives to the 1-D type formula and make use of the advantages of 1-D type formula in multidimensions? Taking a closer look at the LS matrices, we observe that these are real and symmetric matrices with real eigenvalues and a real and distinct set of eigenvectors. The eigenvectors of these matrices are orthogonal. Along the eigendirections, the corresponding LS formulae reduce to the 1-D type formulae. But a problem now arises in combining the eigendirections along with upwinding. Upwinding, which in LS is done by stencil splitting, is essential to provide stability to the numerical scheme. It involves choosing a direction for enforcing upwinding. The stencil is split along the chosen direction. But it is not necessary that the chosen direction is along one of the eigendirections of the split stencil. Thus in general we will not be able to use the 1-D type formulae along the chosen direction. This difficulty has been overcome by the use of weights leading to WLSKUM-ED (Weighted Least Squares Kinetic Upwind Method using Eigendirections). In WLSKUM-ED weights are suitably chosen so that a chosen direction becomes an eigendirection of A(w). As a result, the multi-dimensional LS formulae reduce to 1-D type formulae along the eigendirections. All the advantages of the 1-D LS formuale can thus be made use of even in multi-dimensions. A very simple and novel way to calculate the positive weights, utilizing the coordinate differentials of the neighbouring nodes in the connectivity in 2-D and 3-D, has been developed for the purpose. This method is based on the fact that the summations of the coordinate differentials are of different signs (+ or -) in different quadrants or octants of the split stencil. It is shown that choice of suitable weights is equivalent to a suitable decomposition of vector space. The weights chosen either fully diagonalize the least squares matrix ie. decomposing the 3D vector space R3 as R3 = e1 + e2 + e3, where e1, e2and e3are the eigenvectors of A (w) or the weights make the chosen direction the eigendirection ie. decomposing the 3D vector space R3 as R3 = e1 + ( 2-D vector space R2). The positive weights not only prevent the denominator of the 1-D type LS formulae from going to zero, but also preserve the LED property of the least squares method. The WLSKUM-ED has been successfully applied to a large number of 2-D and 3-D test cases in various flow regimes for a variety of point distributions ranging from a simple cloud generated from a structured grid generator (shock reflection problem in 2-D and the supersonic flow past hemisphere in 3-D) to the multiple chimera clouds generated from multiple overlapping meshes (BI-NACA test case in 2-D and FAME cloud for M165 configuration in 3-D) thus demonstrating the robustness of the WLSKUM-ED solver. It must be noted that the second order acccurate computations using this method have been performed without the use of the limiters in all the flow regimes. No spurious oscillations and wiggles in the captured shocks have been observed, indicating the preservation of the LED property of the method even for 2ndorder accurate computations. The convergence acceleration of the WLSKUM-ED code has been achieved by the use of LUSGS method. The use of 1-D type formulae has simplified the application of LUSGS method in the grid-free framework. The advantage of the LUSGS method is that the evaluation and storage of the jacobian matrices can be eliminated by approximating the split flux jacobians in the implicit operator itself. Numerical results reveal the attainment of a speed up of four by using the LUSGS method as compared to the explicit time marching method. The 2-D WLSKUM-ED code has also been used to perform the internal flow computations. The internal flows are the flows which are confined within the boundaries. The inflow and the outflow boundaries have a significant effect on these flows. The accurate treatment of these boundary conditions is essential particularly if the flow condition at the outflow boundary is subsonic or transonic. The Kinetic Periodic Boundary Condition (KPBC) which has been developed to enable the single-passage (SP) flow computations to be performed in place of the multi-passage (MP) flow computations, utilizes the moment method strategy. The state update formula for the points at the periodic boundaries is identical to the state update formula for the interior points and can be easily extended to second order accuracy like the interior points. Numerical results have shown the successful reproduction of the MP flow computation results using the SP flow computations by the use of KPBC. The inflow and the outflow boundary conditions at the respective boundaries have been enforced by the use of Kinetic Outer Boundary Condition (KOBC). These boundary conditions have been validated by performing the flow computations for the 3rdtest case of the 4thstandard blade configuration of the turbine blade. The numerical results show a good comparison with the experimental results.

The numerics of hyperbolic conservation laws, e.g., the Euler equations of gas dynamics, shallow water equations and MHD equations, is non-trivial due to the convective terms being highly non-linear 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 Lax-Friedrichs (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 Upwind-Biased 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 co-efficient follows a new format unlike the use of maximum characteristic speed in the Local Lax-Friedrichs method and the scheme results in improved solutions of the flow features. The grid-aligned steady contacts are captured exactly with the reorganized format of diffusion co-efficient. 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 Rankine-Hugoniot Solver, attempts to integrate more physics into the discretization procedure by enforcing a simplified Rankine-Hugoniot 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 (MOVERS-1), multiple waves (MOVERS-n) and limiter based diffusion (MOVERS-L) are presented. The scheme is demonstrated for scalar Burgers equation and systems of conservation laws like Euler equations, ideal Magneto-hydrodynamics equations and shallow water equations. The new scheme uniformly improves the solutions of the Local Lax-Friedrichs scheme on which it is based and captures steady discontinuities either exactly or very accurately. A Grid-Free Central Solver, which does not require a grid structure but operates on any random distribution of points, is presented. The grid-free scheme is generic in discretization of spatial derivatives with the location of the mid-point between a point and its neighbor being used to define a relevant coefficient of numerical dissipation. A new central scheme based on convective-pressure splitting to solve for mid-point flux is proposed and many test problems are solved effectively. The Rankine-Hugoniot Solver, which is developed in this thesis, is also implemented in the grid-free framework and its utility is demonstrated. The numerical methods presented are solved in a finite volume framework, except for the Grid-Free 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 eigen-structure 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.