NUMERICAL STUDY OF HYPERBOLIC CONSERVATION LAWS USING FUZZY FLUX LIMITERS

Abstract

Numerous engineering applications rely on a detailed understanding of fluid flows. These applications cross a broad range of industries including aerospace, en ergy, medical, and automotive. Therefore, an understanding into the fundamen tals of fluid dynamics has the potential to impact a broad range of industries, having a tremendously positive impact on society. In Computational Fluid Dy namics (CFD), the development of high-order algorithms is still a very active area of research. These techniques have the ability to lower the computational cost of computing solutions to a desirable precision and accuracy. This thesis aims to develop a high order finite volume (FV) flux limited scheme for hyperbolic con servation laws (HCLs). The proposed technique achieves a high order spatial reconstruction, blended with an optimized switching to limited piece-wise linear reconstruction in uneven regions to preserve the monotonic property. Addition ally, for time marching, the fourth order Runge Kutta (RK) approach is applied. The proposed methodology has been tested using a combination of high-order functional reconstructions and various popular test problems such as the system of Euler equations. Various experiments have been performed to show case the suitability and the use of the modified technique. The main target of this thesis is to explore the numerical approximation of solu tions to some nonlinear conservation laws (CLs). The common basis is the fact that we have some underlying uni-dimensional scalar nonlinear equations which are perturbed in some nonlinear way. Nonlinear hyperbolic equations have been studied for many years because they can be applied in a large number of disci plines. Indeed, many problems in fluid- dynamics, elasticity, biology, mechanics and in each case where it is possible to model the macroscopic evolution of cer tain quantities in time, involve the study of nonlinear hyperbolic equations. The main focus has been for many years on fluid and gas dynamics. Starting from the book of C.B. Laney [22] and from the studies of R.J Leveque [95] there was a considerable development of theoretical studies of well-posedness for nonlinear waves and a great deal has been done in particular for the numerical approxi mation. The main difficulties consist in constructing a numerical method which is stable and preserves consistency. In other words a convergent scheme, which keeps a good accuracy in reconstructing the solution. The principal approach is to start from the ideas of the linear case. The work is concentrated towards the following purposes: • to construct high order numerical schemes; • to obtain hybrid schemes, that is to say, numerical methods by collaborating two entirely different subjects. As we will see, this work combines HCLs and Fuzzy Logic (FL) domains to de velop optimization in the existing numerical techniques for approximating conser vation laws (CLs). The thesis entitled “Numerical Study of Hyperbolic Conservation Laws Using Fuzzy Flux Limiters” includes six chapters followed by a conclusion and future scope and references. The bibliography and list of publications have been given at the end of the thesis. The introduction at beginning of each chapter gives a brief outline of the work presented in that chapter. The reported work is presented in the following manner: Chapter 1 titled, “Introduction and Literature Survey”, provides a summary of the related literature as well as an introduction to the mathematical concepts used. This introductory chapter presents a precise review of the hyperbolic problems. The target of the chapter is basically to describe the rudiments of the HCLs. It also sheds light on various existing numerical methods and related terminologies. We Plan to work on problems associated with HCLs, so the focus will be chiefly but not restricted to the above stated problems. The content of this chapter lay down the map and presents the motivation behind the work carried out in the this thesis. In Chapter 2, we investigate some high-order numerical methods relying on fi nite volume (FV) framework for solving problems related to HCLs with an intention to carry out several new methods which would be more realistic and reliable. A few numerical approximations have also been performed to verify the feasibility of the discussed schemes. The results of this chapter are in the form of research xiv paper entitled "Numerical Simulation of Hyperbolic Conservation Laws Us ing High Resolution Schemes with the Indulgence of Fuzzy Logic" published in Lecture Notes in Computer Science, LNCS, Springer, Cham. and in a commu nicated paper entitled "A Comparative Study of High Resolution Methods for Nonlinear Hyperbolic Problems". Chapter 3 is about the reconstruction of several existing FV numerical methods for tackling discontinuities occurring in the HCLs by utilizing the fuzzy inference systems (FISs). In this chapter, a brief discussion regarding the working mech anism of FIS has been included. Also, this chapter describes the algorithm to construct flux limiting methods using fuzzy tools. The theoretical findings along with the experimental activity ensures the importance of the discussed numerical techniques. The content of this chapter is in the form of research paper entitled "An improved flux limiter using fuzzy modifiers for Hyperbolic Conservation Laws" published in the journal Mathematics and Computers in Simulation (Else vier). Through Chapter 4, a hybrid approach is presented which leads to the hybrid fuzzy flux limiting schemes. This chapter discusses about the optimized flux limited scheme for numerical simulations of HCLs by applying FL-based operator functions. The construction process of the numerical scheme is explored through the possibility and benefit of solving a conservation law using a fuzzy modifier function with a suitable intensity concerning the type of initial data. For laying a background, the essential concepts such as fuzzy modifier functions have been covered. Illustrations of the optimized numerical schemes have been shown here. The outcomes of this chapter are published as research paper entitled "A New Reconstruction of Numerical Fluxes for Conservation Laws using Fuzzy Op erators”, International Journal for Numerical Methods in Fluids (Wiley). In Chapter 5, we will try to extend the concept of optimized schemes to multi dimensions. To be specific this chapter explores the use of Monotone Upstream Schemes for Conservation Laws (MUSCL), in particular flux limiting techniques based on the piecewise polynomial reconstruction for the approximation of bi dimensional Euler system of equations bin fluid dynamics. The numerical frame work corresponding to the hybrid flux limiting techniques in two dimensions has been described in detail. To study the non-oscillatory behaviour of the method and the resolution of discontinuities, the corresponding Riemann problem is ap proximated numerically. The computations are performed on structured meshes. xv Further, for investigating the efficiency and accuracy of the optimized schemes, various numerical illustrations based on the two dimensional Euler system of equations have been carried out. The content of this chapter is in the form of communicated research work entitled "Fuzzy Flux Limiting High-Order Discon tinuity Capturing Scheme for Two Dimensional Euler Equations". Chapter 6 is mainly about the numerical validation of the fuzzified HR hybrid schemes, discussed in the previous chapters. We intend to use several popular real life models to investigate the efficiency and reliability of the computed solu tions obtained by using the fuzzy flux limiters for these problems. Also we would be considering different test problems to check the behaviour of reconstructed and the newly designed methods. Error analysis are performed on the system with the numerical simulation process being done at the end. The content of the chapter is published as research paper entitled "An improved flux limiter using fuzzy modifiers for Hyperbolic Conservation Laws" published in the journal Math ematics and Computers in Simulation (Elsevier), another publisher paper titled "High Resolution TVD Scheme based on Fuzzy Modifiers for Shallow-Water equations", Lecture Notes in Computer Science, LNCS, Springer, Cham., and a communicated research work entitled "On the Approximation of Dam-Break Problems Using a Fuzzified HR-TVD Scheme" Subsequently, the conclusion of the work carried out in the thesis is presented. We have also discussed the future scope of the current work.