Cossey L. Introduction to Mathematical Physics

Подождите немного. Документ загружается.

1. Differential equation is written as:

1. The solution u

of the problem when ε is zero is known.

2. General solution is seeked as:

1. Function N(u) is developed around u

using Taylor type formula:

1. A hierarchy of linear equations to solve is obtained:

This method is simple but singular problem my arise for which solution is not valid

uniformly in t.

Example:

Non uniformity of regular perturbative expansions

Remark:

Origin of secular terms : A regular perturbative expansion of a periodical function whose

period depends on a parameter gives rise automatically to secular terms

Born's iterative method

Algorithm:

1. Differential equation is transformed into an integral equation:

1. A sequence of functions u

converging to the solution u is seeked:

Starting from chosen solution u

, successive u

are evaluated using recurrence formula:

This method is more "global" than previous one \index{Born iterative method} and can

thus suppress some divergencies. It is used in diffusion problems

). It has the drawback to allow less

control on approximations.

Multiple scales method

Algorithm:

1. Assume the system can be written as:

1. Solution u is looked for as:

with T = ε t

for all .

1. A hierarchy of equations to solve is obtained by substituting

expansion eqdevmu into equation eqavece.

For examples see ).

Poincar\'e-Lindstedt method

This method is closely related to previous one, but is specially dedicated to studying

periodical solutions. Problem to solve should be:\index{Poincar\'e-Lindstedt}

Problem:

Find u such that:

LABELeqarespoG(u,ω) = 0

where u is a periodic function of pulsation ω. Setting τ = ωt, one gets:

u(x,τ + 2π) = u(x,τ)

Resolution method is the following:

Algorithm:

1. Existence of a solution u (x)

which does not depend on τ is

imposed:

G(u (x),ω) = 0LABELfix

Solutions are seeked as:

with u(x,τ,ε = 0) = u (x)

1. A hierarchy of linear equations to solve is obtained by expending G around u

and

substituting form1 and form2 into eqarespo.

==WKB method==LABEL mathsecWKB

WKB (Wentzel-Krammers-Brillouin) method is also a perturbation

method. It will be presented at section

secWKB in the proof of ikonal equation.

Evolution, numerical time integration

Introduction

There exist very good books )) about numerical integration of PDE evolution problem.

Note that all methods (finite difference, finite element, spectral methods) contains as a

final step the time integration of a ODE system. In this section this time integration is

treated. Problem to be solved is the following

Problem: LABEL probmathevovec (Cauchy problem) Find functions u (t)

obeying the

ODE system

where u (x,t = 0)

is known.

We will not present here details about stability and precision calculation of the numerical

integration schemes, however the reader should always keep in mind this crucial

problem. Generally speaking, knowledge of mathematical properties of solutions should

always be used to verify numerical results. For instance, if the solution of problem

probmathevovec is known to be bounded, and that the solution obtained

numerically diverges, then numerical solution will be obviously considered as bad. This

problem of stability is the first that the numerician meet. However, more refined

considerations have to be considered. Integrals of movement are a classical way to check

accuracy of solutions. For hamiltonian systems for instance, energy conservation should

be checked at regular time intervals.

Euler method

Euler method consists in approximating the time derivative

However, the way the right-hand term is evaluated is very important for the stability of

the integration scheme

). So, explicit scheme:

is unstable if . On another hand, implicit scheme:

is stable. This last scheme is called "implicit" because equation eqimpli have to be solved

at each time step. If F is linear, this implies to solve a linear system of equations

)).

One can show that the so-called Cranck-Nicholson scheme:

is second order in time. Another interesting scheme is the leap-frog scheme:

Example:

Let us illustrate Euler method for solving Van der Pol equation:

which can also be written:

Example:

Let us consider the reaction diffusion system:

Figure figreacd shows results of integration of this system using Euler method (space

treatment is achieved using simple finite differences, periodic boundary conditions are

used). The lattice has sites.

Example:

Nonlinear Klein-Gordon equation

u − u = F(u)

xx tt

where F is (Sine-Gordon equation)

F(u) = sinu

or (phi four)

F(u) = − u + u

can be integrated using a leap-frog scheme

Runge-Kutta method

Runge-Kutta formula \index{Runge-Kutta} at fourth order gives u

i + 1

as a function of y

where

Adams method (multiple steps)

Problem:

Find u(t) solution of:

with

u(0) = u

can be solved by open Adams\index{Adams formula} method order two:

or open Adams method order three:

Open Adams method are multisteps: they can not initiate an integration process.

Examples of closed Adams method, are at order two:

and order three:

Closed Adams methods are also multisteps. They are more accurate but are implicit: they

imply the solving of equations systems at each time step. Predictor--corrector method

provides a practical way to use closed Adams formulas.

Predictor--corrector method

Predictor--corrector methods\index{predictor-corrector} are powerful methods very

useful in the case where right--hand side is complex. An first estimation is done

(predictor) using open Adams formula, for instance open Adams formula order three:

Then, u

i + 1

is re-evaluated using closed Adams formula where computed previously

is used to avoid implicity:

Remark:

Evaluation of F in the second step is stabilizing. Problem

where L is a linear operator and N a nonlinear operator will be advantageously integrated

using following predictor--corrector scheme:

Symplectic transformation case

Consider\index{symplectic} the following particular case of evolution problem:

Problem:

Find solution [q(t),p(t)]

where [q(t = 0),p(t = 0)] is known (Cauchy problem) and where functions f and g come

from a hamiltonian H(q,p):

Dynamics in this case preserves volume element dq.dp. Let us define symplectic

\index{symplectic}transformation:

Definition:

Transformation T defined by:

is symplectic if its Jacobian has a determinant equal to one.

Remark:

Euler method adapted to case:

consists in integrating using the following scheme:

and is not symplectic as it can be easily checked.

Let us giev some examples of symplectic integrating schemes for system of equation

eqsymple1 and eqsymple2.

Example:

Verlet method:

is symplectic (order 1).

Example:

Staggered leap--frog scheme

is symplectic (order 2).

Figures codefigure1, codefigure2, and codefigure3 represent trajectories computed by

three methods: Euler, Verlet, and staggered leap--frog method for (linear) system

describing harmonic oscillator:

Particular trajectories and geometry in

space phase

Fixed points and Hartman theorem

Consider the following initial value problem:

with x(0) = 0. It defines a flow: defined by φ (x ) = x(t,x )

t 0 0

By Linearization around a fixed point such that :