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Discontinuous Galerkin Finite Element Method for The DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR THE NONLINEAR HYPERBOLIC PROBLEMS WITH ENTROPY-BASED ARTIFICIAL VISCOSITY STABILIZATION A Dissertation by VALENTIN NIKOLAEVICH ZINGAN Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of Doctor of Philosophy May 2012 Major Subject: Nuclear Engineering DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR THE NONLINEAR HYPERBOLIC PROBLEMS WITH ENTROPY-BASED ARTIFICIAL VISCOSITY STABILIZATION A Dissertation by VALENTIN NIKOLAEVICH ZINGAN Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Approved by: Co-Chairs of Committee, Jim E. Morel Jean-Luc Guermond Committee Members, Marvin L. Adams Jean C. Ragusa Pavel V. Tsvetkov Head of Department, Yassin A. Hassan May 2012 Major Subject: Nuclear Engineering iii ABSTRACT Discontinuous Galerkin Finite Element Method for the Nonlinear Hyperbolic Problems with Entropy-Based Artificial Viscosity Stabilization. (May 2012 ) Valentin Nikolaevich Zingan, B.S.N.E., Moscow State Engineering Physics Institute (MEPhI); M.S.N.E., Moscow State Engineering Physics Institute (MEPhI) Co-Chairs of Committee: Dr. Jim E. Morel Dr. Jean-Luc Guermond This work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabi- lization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p > 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultane- ously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the lo- cal size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. iv One- and two-dimensional benchmark test cases are presented for nonlinear hy- perbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and op- timal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization. v DEDICATION To my father Nikolay and my mother Svetlana vi ACKNOWLEDGMENTS This research project would not have been possible without the support of many people. First of all, I wish to express my gratitude to my advisor, Prof. Jim E. Morel who was abundantly helpful and offered invaluable assistance, support and guidance. I would like to sincerely thank my co-advisor Prof. Jean-Luc Guermond for giving me valuable ideas and getting me familiar with the world of finite elements. Prof. Bojan Popov is gratefully acknowledged for his collaboration and useful suggestions. I would like to express my special thanks to the deal.II community, especially Prof. Wolfgang Bangerth. I always received the valuable replies to all my requests. I am forever indebted to my father Nikolay Zingan and my mother Svetlana Zingan for their love, understanding, endless patience and encouragement when it was most required. Finally, yet importantly, I would like to express my heartfelt thanks to my beloved wife Alena and my little son Daniel for their love, support and encouragement during the entire period of this work. vii NOMENCLATURE BC’s boundary conditions BDF backward differentiation formula CD centered difference CFD computational fluid dynamics CFL Courant-Friedrichs-Lewy CG continuous Galerkin CN Crank-Nicholson DG discontinuous Galerkin DoF degree of freedom ENO essentially non-oscillatory EoS equationofstate FE finiteelement FEM finite element method FVM finite volume method IBVP initial boundary value problem IVP initial value problem JLG Jean-Luc Guermond KPP Kurganov-Petrov-Popov LES large eddy simulation LF Lax-Friedrichs LTE local truncation error ODE ordinary differential equation PDE partial differential equation RK Runge-Kutta TV totalvariation TVD total variation diminishing viii WENO weighted essentially non-oscillatory ix TABLE OF CONTENTS Page ABSTRACT .................................... iii DEDICATION ................................... v ACKNOWLEDGMENTS ............................. vi NOMENCLATURE ................................ vii TABLE OF CONTENTS ............................. ix LIST OF TABLES ................................. xii LIST OF FIGURES ................................ xiv 1. INTRODUCTION ............................... 1 1.1 Motivation ................................. 1 1.2 Background ................................ 2 1.2.1 Governing equations ....................... 2 1.2.2 Higher-order numerical methods ................. 3 1.2.3 Shock capturing .......................... 5 1.2.4 Artificial viscosities ........................ 7 1.3 Thesis overview .............................. 9 2. CONSERVATION LAWS ........................... 10 2.1 Hyperbolic scalar conservation laws ................... 10 2.1.1 Integral and differential forms .................. 10 2.1.2 Diffusion .............................. 13 2.1.3 Shock formation .......................... 14 2.1.4 Vanishing viscosity solution ................... 19 2.1.5 Weak solution ........................... 21 2.1.6 One-dimensional Riemann problem as an example of nonunique- ness of a weak solution ...................... 22 2.1.7 Entropy conditions ........................ 26 2.2 Foundations of fluid dynamics ...................... 31 2.2.1 Compressible Navier-Stokes equations .............. 31 2.2.2 Viscosity and thermal conduction ................ 32 2.2.3 Compressible Euler equations .................. 33 x 2.2.4 Equilibrium thermodynamics of a perfect gas .......... 34 2.2.5 Entropy .............................. 38 3. THE IDEA OF NUMERICAL STABILIZATION AND ARTIFICIAL VIS- COSITIES .................................... 42 3.1 Finite difference approximation ..................... 42 3.1.1 Linear transport equation .................... 42 3.1.2 Extension to nonlinear scalar equations ............. 54 3.1.3 Extension to multi-dimensional grids .............. 56 3.1.4 Extension to nonuniform grids .................. 56 3.2 DG(0) approximation and numerical fluxes ............... 57 3.3 Entropy-based artificial viscosity ..................... 64 4. DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD AND ENTROPY-BASED ARTIFICIAL VISCOSITY APPROXIMATION FOR SOLVING HYPERBOLIC SCALAR CONSERVATION LAWS ....... 67 4.1 Introduction ................................ 67 4.2 Preface ................................... 67 4.3 Model problem .............................. 70 4.4 Discretization in space .......................... 71 4.4.1 Mesh and some notation ..................... 71 4.4.2 Standard DG formulation .................... 73 4.4.3 Numerical fluxes ......................... 76 4.5 Discretization in time ........................... 81 4.6 Inclusion of the entropy-based artificial viscosity ............ 85 4.7 Numerical tests .............................. 88 4.7.1 One-dimensional tests ...................... 88 4.7.2 Two-dimensional tests ...................... 101 4.8 Conclusion ................................. 110 5. EXTENSION OF THE METHOD TO COMPRESSIBLE GAS DYNAM- ICS EQUATIONS ................................ 111 5.1 Introduction ................................ 111 5.2 Regularization of Euler equations .................... 111 5.3 Discretization in space .......................... 113 5.3.1 Inviscid numerical fluxes ..................... 115 5.3.2 Viscous numerical fluxes ..................... 116 5.4 Discretization in time ........................... 117 5.5 Artificial viscosities ............................ 119 5.6 Numerical tests .............................. 121 5.6.1 One-dimensional tests ...................... 121 5.6.2 Two-dimensional tests ...................... 127 xi 6. CONCLUSIONS ................................ 136 REFERENCES ................................... 138 xii LIST OF TABLES TABLE Page 4.1 General Butcher tableau. ........................... 84 4.2 Explicit RK3 (left) and RK4 (right) Butcher tableaux. .......... 84 4.3 1D linear transport equation with smooth data. Parameters. ....... 88 4.4 1D linear transport equation with smooth data. Convergence test for P1. 89 4.5 1D linear transport equation with smooth data. Convergence test for P2. 90 4.6 1D linear transport equation with smooth data. Convergence test for P3. 90 4.7 1D linear transport equation with discontinuous data. Parameters. ... 92 4.8 1D linear transport equation with discontinuous data. Convergence test for P1. ..................................... 93 4.9 1D linear transport equation with discontinuous
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