The method of or4er reduction has been developed on the basis of the well-known Keller's box scheme. It is an indirect method of constructing difference schemes for approximating the differential equations. First, some new variables are introduced for the reduction of the original problem into an equivalent system of lower order differential equations and a difference scheme is constructed for the latter. Then, the discrete variables are separated to obtain a difference scheme only containing the original variables.

The layout of this book is as follows. Chapter 1 provides a microcosm of the method of order reduction via a two-point boundary value problem. Chapters 2, 3 and 4 are devoted, respectively, to the numerical solutions of linear parabolic, hyperbolic and elliptic equations by the method of order reduction. They are the core of the book. Chapters 5, 6 and 7 respectively consider the numerical approaches to the heat equation with an inner boundary condition, the heat equation with a nonlinear boundary condition and the nonlocal parabolic equation. Chapter 8 discusses the numerical approximation to a fractional diffusion-wave equation. The next five chapters are devoted to the numerical solutions of several coupled systems of differential equations. The numerical procedures for the heat equation and the Burgers equation in the unbounded domains are studied in Chapters 14, 15 and 16. Chapter 17 provides a numerical method for the superthermal electron transport equation, which is a degenerate and nonlocal evolutionary equation. The numerical solution to a model in oil deposit on a moving boundary is presented in Chapter 18. Chapter 19 deals with the numerical solution to the Cahn-Hilliard equation, which is a fourth order nonlinear evolutionary equation. The ADI and compact ADI methods for the multidimensional parabolic problems are discussed in Chapter 20. The numerical errors in the maximum norm are obtained. Chapter 21, the last chapter, is devoted to the numerical solution to the time-dependent SchrSdinger equation in quantum mechanics.

Chapter 1 The Method of Order Reduction

 1.1 Introduction

 1.2 First order off-center difference method

 1.3 Second order off-center difference method

 1.4 Method of fictitious domain

 1.5 Method of order reduction

 1.6 Comparisons of the four difference methods

 1.7 Conclusion

Chapter 2 Linear Parabolic Equations

 2.1 Introduction

 2.2 Derivative boundary conditions

 2.3 Derivation of the difference scheme

 2.4 A priori estimate for the difference solution

 2.5 Solvability, stability and convergence

 2.6 Two dimensional parabolic equations

 2.7 Conclusion

Chapter 3 Linear Hyperbolic Equations

 3.1 Introduction

 3.2 Derivation of the difference scheme

 3.3 A priori estimate

 3.4 Solvability, stability and convergence

 3.5 Numerical examples

 3.6 Conclusion

Chapter 4 Linear Elliptic Equations

 4.1 Introduction

 4.2 Derivation of the difference scheme

 4.3 Solvability, stability and convergence

 4.4 The Neumann boundary value problem

 4.5 A numerical example

 4.6 Conclusion

Chapter 5 Heat Equations with an Inner Boundary Condition

 5.1 Introduction

 5.2 Derivation of the difference scheme

 5.3 Solvability, stability and convergence

 5.4 A numerical example

 5.5 Conclusion

Chapter 6 Heat Equations with a Nonlinear Boundary Condition

 6.1 Introduction

 6.2 Derivation of the difference scheme

 6.3 Convergence of the difference scheme

 6.4 Unique solvability of the difference scheme

 6.5 Iterative algorithm and a numerical example

 6.6 Conclusion

Chapter 7 Nonlocal Parabolic Equations

 7.1 Introduction

 7.2 Derivation of the difference scheme

 7.3 A prior estimate

 7.4 Convergence and solvability

 7.5 Extrapolation method

 7.6 Implementation of the difference scheme

 7.7 Conclusion

Chapter 8 Fractional Diffusion-wave Equations

 8.1 Introduction

 8.2 Approximation of the fractional order derivatives

 8.3 Derivation of the difference scheme

 8.4 Analysis of the difference scheme

 8.5 A compact difference scheme

 8.6 A slow diffusion system

 8.7 A numerical example

 8.8 Conclusion

Chapter 9 Wave Equations with Heat Conduction

 9.1 Introduction

 9.2 Boundary conditions

 9.3 Derivation of the difference scheme

 9.4 Solvability, stability and convergence

 9.5 A practical recurrence algorithm

 9.6 The degenerate problem

 9.7 Conclusion

Chapter 10 Timoshenko Beam Equations with Boundary Feedback

 10.1 Introduction

 10.2 Derivation of the difference scheme

 10.3 Analysis of the difference scheme

 10.4 A numerical example

 10.5 Conclusion

Chapter 11 Thermoplastic Problems with Unilateral Constraint

 11.1 Introduction

 11.2 Derivation of the difference scheme

 11.3 Stability and convergence

 11.4 Numerical examples

 11.5 Conclusion

Chapter 12 Thermoelastic Problems with Two-rod Contact

 12.1 Introduction

 12.2 Derivation of the difference scheme

 12.3 Stability and convergence

 12.4 Solvability and iterative algorithm

 12.5 Numerical examples

 12.6 Conclusion

Chapter 13 Nonlinear Parabolic Systems

 13.1 Introduction

 13.2 Difference scheme

 13.3 Unique solvability and convergence

 13.4 A numerical example

 13.5 Conclusion

Chapter 14 Heat Equations in Unbounded Domains

 14.1 Introduction

 14.2 Derivation of the difference scheme

 14.3 Analysis of the difference scheme

 14.4 A numerical example

 14.5 Conclusion

Chapter 15 Heat Equations on a Long Strip

 15.1 Introduction

 15.2 Derivation of the difference scheme

 15.3 Analysis of the difference scheme

 15.4 A numerical example

 15.5 Conclusion

Chapter 16 Burgers Equations in Unbounded Domains

 16.1 Introduction

 16.2 Reformulation of the problem

 16.3 Derivation of the difference scheme

 16.4 Solvability and stability of the difference scheme

 16.5 Convergence of the difference scheme

 16.6 A numerical example

 16.7 Conclusion

Chapter 17 Superthermal Electron Transport Equations

 17.1 Introduction

 17.2 Derivation of the difference scheme

 17.3 Analysis of the difference scheme

 17.4 A numerical example

 17.5 Conclusion

Chapter 18 A Model in Oil Deposit

 18.1 Introduction

 18.2 Difference scheme and the main results

 18.3 Derivation of the difference scheme

 18.4 Solvability and convergence

 18.5 Conclusion

Chapter 19 The Two-dimensional Cahn-Hillard Equation

 19.1 Introduction

 19.2 Derivation of the difference scheme

 19.3 Solvability and convergence of the difference scheme

 19.4 Conclusion

Chapter 20 ADI and Compact ADI Methods

 20.1 Introduction

 20.2 Notations and auxiliary lemmas

 20.3 Error analysis of the ADI solution and its extrapolation

 20.4 Error estimates of the compact ADI method

 20.5 A numerical example

 20.6 Conclusion

Chapter 21 Time-dependent SchrSdinger Equations

 21.1 Introduction

 21.2 One-dimensional Crank-Nicolson scheme

 21.3 An extension to the high-order compact scheme

 21.4 Extensions to multidimensional problems

 21.5 Treatment of the nonhomogeneous boundary conditions

 21.6 A numerical example

 21.7 Conclusion

Bibliography