NUMERICAL SIMULATIONRESEARCH PROGRESS -- CONTENTS -- PREFACE -- THE APPLICATION OF THE METHOD OFCHARACTERISTICS FOR THE NUMERICAL SOLUTIONOF HYPERBOLIC DIFFERENTIAL EQUATIONS -- Abstract -- 1. Introduction -- 2. The Collisionless Kinetic Sheath -- 2.1. The Relevant Equations -- 2.2. The Numerical Scheme -- 2.3. Results -- 3. Study of the Phase-Space Dynamic in Capacitive Discharges -- 4. A One-Dimensional Ion Extraction Model -- 5. Oscillations of the Collisionless Sheath at Grazing Incidence ofthe Magnetic Field -- 5.1. The Kinetic Model for the Magnetized Sheath -- 5.2. The Numerical Scheme -- 6. Study of the Formation of a Charge Separation and an ElectricField at a Plasma Edge -- 6.1. The Relevant Equations and the Numerical Method for the 2D Problemin Cylindrical Geometry -- 6.2. Results -- Case1 -- Case2 -- 7. Numerical Simulation of Wake-Field Acceleration -- 7.1. The Relevant Equations -- The 1D relativistic Vlasov-Maxwell model -- The numerical scheme -- 7.2. Results -- The case of a circular polarization -- The case of a linear polarization -- 8. Interaction of a High Intensity Laser Field Incident on anOverdense Plasma -- 9. Fuid Equations -- 9.1. A One-Dimensional Model for the Blood Flow in the Aorta -- 9.2. Acoustic Waves -- 10. Conclusion -- Acknowledgments -- References -- MIXED FINITE DIFFERENCE-SPECTRALNUMERICAL APPROACH FOR KINETIC AND FLUIDDESCRIPTION OF NONLINEAR PHENOMENA INPLASMA PHYSICS -- Abstract -- 1. Introduction -- 2. Kinetic Point of View in Plasma Physics -- 2.1. Hyperbolic Equations of Conservation Law Type -- 2.2. Splitting Method -- 3. The Fluid Point of View -- 3.1. The Magnetohydrodynamics Approximation -- 3.2. Numerical Solution of the MHD Equations -- 3.3. Advection Equations -- 3.4. Elliptic Equations -- 3.5. Boundary Conditions for the MHD Description -- 4. Kinetic Simulations -- 4.1. Vlasov-Poisson Code.

4.2. Linear and Nonlinear Landau Damping -- 4.3. PlasmaWaves Echoes -- 4.4. Phase Space Vortex Coalescence -- 5. Magnetohydrodynamics Simulations -- 6. Conclusions -- References -- NUMERICAL SIMULATIONS OF THE NONLINEARSOLITARY WAVES -- Abstract -- 1. Introduction -- 2. The Symlectic and Multisymplectic Methods -- 3. Simulations of SolitaryWaves by Symplectic Methods -- 3.1. Simulations of the Coupled Nonlinear Schrödinger System -- 3.2. Simulations of the Nonlinear Rossby Wave Packets -- 4. Simulations of SolitaryWaves by Multi-Symplectic Methods -- 4.1. Simulations of the Nonlinear Klein-Gordon Equation -- 4.2. Simulations of the Kdv Equation -- 5. Conclusion -- Acknowledgements -- References -- SYMMETRY IN TURBULENCE SIMULATION -- Abstract -- 1. Introduction -- 2. Panorama of the Application of Symmetries -- 2.1. Basic Definitions -- 2.2. Resolution of a Riccati Equation -- 2.3. Integrating Factor -- 2.4. Reduction of a Partial Differential Equation -- 2.5. 2D Laminar Thin Shear Layer Flows -- 2.5.1. Scaling Symmetries and Self-similar Solutions -- 2.5.2. Reduction of the Equations -- 2.5.3. Examples of Values of -- 2.6. Non-isothermal Laminar Thin Shear Layer Flows -- 2.7. Burger's Vortex and Shear Layer Solutions of the Navier-Stokes Equations -- 3. Computation of One-Parameter Symmetries -- 4. Symmetry in Turbulence Modeling -- 4.1. Isothermal Navier-Stokes Equations -- 4.2. Turbulence Model Analysis -- 4.3. Example of Symmetry-Preserving Turbulence Models -- 4.4. Consequences of the Second Law of Thermodynamics -- 5. Numerical Test -- 5.1. Non-isothermal Flow -- 5.1.1. Model Analysis -- 5.1.2. New Symmetry-Preserving Turbulent Models -- 6. Invariant Schemes -- 6.1. Basic Definitions -- 6.2. Invariantization of a Numerical Scheme -- 6.3. Application to the Burgers' Equation -- 6.3.1. Transformation of the Grid.

6.3.2. Invariantization of the Scheme -- 6.3.3. Determination of a4 and a5 -- 6.3.4. Order of Accuracy -- 6.4. Numerical Tests -- 7. Conclusion -- References -- THE SHOOTING METHOD IN HYDROTHERMALOPTIMAL CONTROL PROBLEMS -- Abstract -- 1. Introduction -- 2. Problem without Restrictions -- 2.1. Existence and Uniqueness of Extremals -- 2.2. Shooting Mappings -- 2.3. Existence and Uniqueness of Local Solution -- 2.4. A Particular Case -- 2.5. Solutions for Convex Functionals -- 2.6. Solutions for Non-Convex Functionals -- 2.7. Optimization Algorithm -- 3. Problem with Restrictions -- 3.1. Existence of Solution -- 3.2. Interior Solutions -- 3.3. Boundary Solutions -- 3.4. Optimization Algorithm -- 4. Examples -- 4.1. Example 1: A Problem without Restrictions -- 4.2. Example 2: A Problem with Restrictions -- 4.3. Example 3: Fields of Extremals -- 5. Conclusion -- References -- EXACT N-SOLITON SOLUTIONSOF THE SHARMA-TASSO-OLVER-KADOMTSEVPETVIASHVILI(STO-KP)EQUATION -- Abstract -- 1. Introduction -- 2. The Methods -- 2.1. The tanh-coth Method -- 2.2. The Hirota's Bilinear Method -- 3. Using the tanh-coth Method -- 4. Using the Hirota's Bilinear Method -- 5. Conclusion -- References -- ADVANCES IN NUMERICAL SIMULATIONOF GRANULAR MATERIAL -- Abstract -- 1. Granular Material in General -- 2. Numerical Simulation of Granular Material -- 3. Collision Modeling -- 4. Continuum Type Approach -- 4.1. Constitutive Equations in Rate-Independent Quasi-static Regime -- 4.2. Constitutive Equations in Transitional Regime -- 4.3. Viscous-like Behavior -- 4.4. Fluctuation Energy -- 4.5. Continuous Phase -- 4.6. Result of the Continuum Approach in Simulating the Flowof a Vibro-Granular Bed -- 5. Discrete Elements Method -- 5.1. Simulating a Vibro-Granular Bed Using the DEM Approach -- 6. Summary -- References -- INDEX.