Applications of Nonstandard Finite Difference SchemesThe main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter 1 gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain mathematical issues related to the nonstandard discretization of competitive and cooperative models for ecology. Chapter 5 discusses exactness, stability properties, and the symplecticity of various schemes including the conditions for which Runge-Kutta methods are exact. The application chapters illustrate well the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used.This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena. |
Contents
Chapter | 1 |
23 | 8 |
TimeIndependent Schrödinger Equations | 35 |
139 | 45 |
Bibliography | 51 |
AdvectionDiffusionReaction Equations | 64 |
| 106 | |
3 | 124 |
Nonstandard Discretization Methods for Some | 155 |
| 179 | |
| 244 | |
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Common terms and phrases
Advection-Diffusion-Reaction Equations advection-reaction algorithm analytical solution applied approximate solution asymptotically stable backtrack point cm+1 computational constant convergence cubic spline Ď² denominator function derivatives difference equation difference operator discrete model dynamics equa exact finite difference exact scheme FD approximation FDTD finite difference schemes fixed points flip bifurcation given grid point Hamiltonian implicit initial condition initial value integration iterates Lemma linear lintrap logistic growth Maxwell's equations method of characteristics non-standard method nonlinear reaction terms nonstandard finite difference nonstandard schemes NSFD algorithm numerical instabilities numerical scheme numerical solution obtain one-dimensional ordinary differential equations oscillations Partial Differential Equations R. E. Mickens R(cm+¹ reaction term r(c slab solving spline interpolation step step-size symplectic Taylor series Theorem time-stepping scheme tion trajectories transport Eq Uk+1 Un+1 uniform spatial grid unstable velocity field wave equation Yn+1 მე