Numerical Solution of Hyperbolic Differential Equation

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Magdi Mounir Shoucri (Author)

The application of the method of characteristics for the numerical solution of hyperbolic type partial differential equations will be presented. Especial attention will be given to the numerical solution of the Vlasov equation, which is of fundamental importance in the study of the kinetic theory of plasmas, and to other equations pertinent to plasma physics. Examples will be presented with possible combination with fractional step methods in the case of several dimensions. The methods are quite general and can be applied to different equations of hyperbolic type in the field of mathematical physics. Examples for the application of the method of characteristics to fluid equations will be presented, for the numerical solution of the shallow water equations and for the numerical solution of the equations of the incompressible ideal magnetohydrodynamic (MHD) flows in plasmas.

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Table of Contents

Preface

1. Introduction

2. The Fractional Step Method Applied to the Vlasov Equation

2.1. The Fractional Step Method Applied to the Vlasov-Poisson System in One Spatial Dimension

2.2. The Vlasov-Poisson System in Higher Phase-Space Dimensions: The Problem of the Formation of an Electric Field at a Plasma Edge in a Slab Geometry

2.3. Vlasov-Maxwell Equations for Laser-Plasma Interaction

3. Problems Involving the Interpolation along the Characteristic Curves in Two Dimensions

3.1. Solution of the Guiding-Center or Euler Equations

3.2. The Vlasov-Poisson System in Higher Phase-Space Dimensions: Formation of an Electric Field at a Plasma edge in a Cylindrical Geometry

3.3. One-Dimensional Fully Relativistic System for the Problem of Laser-Plasma Interaction

3.4. Numerical Solution of a Reduced Model for the Collisionless Magnetic Reconnection

4. Application of the Method of Characteristics to Fluid Equations

4.1. Numerical Solution of the Shallow Water Equations

4.2. Two-Dimensional Magnetohydrodynamic Flows

5. Conclusion

APPENDIX A The Shift Operator Using the Cubic Spline

APPENDIX B Interpolation Using the Cubic Spline

APPENDIX C Interpolation Using the Cubic B-spline

References

Index

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