CAAM452Lecture8(1)

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Numerical Methods for Partial

Differential Equations

CAAM 452

Spring 2005

Lecture 8

Instructor: Tim Warburton

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CAAM 452 Spring 2005

Recall: Convergence Conditions for

LMM Time-stepping Methods

i. Establish that a unique solution to the ODE exists via Picardstheorem (http://mathworld.wolfram.com/PicardsExistenceTheorem.html)

ii. For time stepping Dahlquists Equivalence Theorem tells us that

a linear multistep time-stepping formula is convergent if and

only if it is consistent and stable

iii. We can easily verify consistency by using Taylor expansions for

the local truncation error.

iv. We check stability conditions by finding roots of the stability

polynomial.

v. A global error analysis tells us that if the right hand side functionis sufficiently smooth (p times continuously differentiable), and

the LMM is stable with local truncation error then the

error at a fixed time converges as

pO dt

1pO dt
http://mathworld.wolfram.com/PicardsExistenceTheorem.htmlhttp://mathworld.wolfram.com/PicardsExistenceTheorem.html

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Consistency of Finite Difference Operator

Definition:

A finite difference operatoris consistent if it converges

towards the continuous operator of the PDE as bothdt0 and dx0

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Example:

Euler-Forward + Right Difference

The finite difference method is:

We define the local truncation error as the operator which

maps the actual solution of the PDE to the correction

required to make it satisfy the scheme at each time step:

1n n

nm mmu u c u

dt

1

2 2*

2

2 2*

2

, ,,

1

, ,2!

, ,2!

n m n m n

m m n

m n m n

m n m n

u x t u x t T u c u x t

dt

u dt u

dt x t x t dt t t

c u dx udx x t x t

dx x x

O dt O dx

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Quick Note

Notice that in the definition of the LTE for the finitedifference scheme we have not multiplied through

by dt (since that would bias the LTE with respect to

dt)

In this example the scheme is said to be first ordermethod accurate in both time and space.

1, , ,n m n m nm m nu x t u x t T u c u x t dt

O dt O dx

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Second Example(Leap Frog in Time and 4th Order Central in Space)

Here we use the fourth order central differencing inspace and Leap Frog in time:

The truncation error in this case is:

Thus we declare the method accuracy to be 2nd

order in time and 4th order in space.

1 1 2

4 0 12 6

n nn nm mm m

u u dxc u c u

dt

1 1

4

2 3 4 5*

5

2

*

3

4

, ,,

2

48, ,6 5!

n m n m n

m m n

m n m n

O

u x t u x t T u c u x t

dt

dt u dx ux t c x tt

O dt d

x

x

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cont

In this case there is a discrepancy between themagnitude of the time stepping error and the spatial

error.

Using this scheme may require a smaller time step

than dx to ensure that the truncation errors for each

part are of similar size.

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Definition: Method Order Accuracy

If the local truncation error satisfies:

Then the method s order accurate in time and rthorder accurate in space.

Again if there is a discrepancy between r and sthen it may be wise to consider reducing dt (if s

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Scheme Notation

For brevity we will denote the linear finite differenceschemes:

where the coefficients may depend on dt,dx as:

then the scheme reads:

1

0

j p j qn j n j

j m j m

j j r

u c u

1

,

0

j p j qn n j n j

dt dx m j m j m

j j r

P u u c u

,0ndt dx mP u

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Stability

Definition:

A finite difference scheme for a first-order PDE is

stable if there is an integer J and positive numbers dt0and dx0

such that for any positive time T, there is a constant CT

such that:

,0ndt dx mP u

1 12 2

0 0 0

0 0

for 0 ,0 and 0

M J Mn j

m T m

m j m

dx u C dx u

ndt T dx dx dt dt

i.e. for a scheme to be stable it must not increase the

solution energy beyond some energy injected at the

start of the time stepping.

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cont (norm notation)

We define a discrete Euclidean norm on the discretesolution as:

Then the stability condition is:

Or equivalently:

1

0

:

or sometimes denoted by

dx

m M

m m

m

h

u dx u u

u

2

0

Jn j

Tdx dxj

u C u

*

0

Jn j

Tdx dxj

u C u

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Well Posedness

Definition:

The initial value problem for a first-order PDE is well

posed if the following holds for all initial data u(x,0)

for some choice of norm (say with integration over

the interval in x ) where the constant C(t) is

independent of the solution.

, ,0u x t C t u x

2L

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Consequences of Well Posedness

If a first order PDE is well posed, it satisfies ananalog of the numerical stability we have been

seeking.

There are two important consequences:

two initial conditions which are almost everywhere

identical will generate two solutions which are almosteverywhere identical.

two solutions which start close together will remain

close togetheralmost everywhere.

, ,0u x t C t u x*

0

Jn j

Tdx dx

j

u C u

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cont

The following PDEs are well-posed

2

2

3

2

2 2

2

u uc su

t x

u usu

t x

u u uc du

t t x x

u u ucu

t t x x

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Convergence

Definition:

A one-step finite difference scheme approximating a PDE

is a convergent scheme if for any solution to the PDE

,u(x,t), with solution to the finite difference scheme, ,such that converges to as m*dx converges tox,

then converges to u(x,t) as (m*dx,n*dt) converges to

(x,t) as (dt,dx) (0,0)

n

mu0

mu 0u x

n

mu

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cont

Convergence requires:

for all solutions

0

0a) if as

b) then , as ,

when 0 0

m

n

m

u u x mdx x

u u x t mdx x ndt t

dt and dx

0, with ,0u x t u x u x

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Lax-Richtmyer Equivalence Theorem

Theorem:

A consistent finite difference scheme for a partial

differential equation for which the initial value problem

is well-posed is convergent if and only if it is stable.

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Accuracy

There is a technical issue in comparing the numericalsolution and the actual solution.

For any specific resolution (choice of dt,dx) the

numerical solution is defined at discrete points inspace and time.

However, the actual solution to the PDE is defined

over the entire interval.

We now discuss how to compare these very different

representations

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Solution Norms

We will need to compare two solutions over theperiodic interval.

We will use conventional L2 and slightly less

conventional Hs,Sobolev, norms:

Notice this Sobolev norm is constructed with

respect to Fourier derivatives

2

2 2

0

22

:

: 1 s

L

L

s

H

u dx u u dx

u dx u

A h 1) C S l ti With

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Approach 1) Compare Solution With

Interpolated Numerical Solution

In the first approach we compare the actual solutionand a trigonometric interpolation of the numerical

solution.

We find a Fourier sum which interpolates the

numerical solution at the M data points.

i.e. we form a Fourier series with uhat coefficients:

21

0

1,

jxMi

n LM n j

j

I u x t u eM

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cont

Where we demand that the interpolant agrees withthe vector of values of the numerical solution

The interpolant is a map from discrete points to a

function defined on the periodic interval, which we

will denote as:

21

0

1,

mjxM in n L

m M m n jj

u I u x t u eM

21 1

0 0

1jxM M

in n L

jk k

j k

Su u e

M

F

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Where Fis the discrete Fourier transform from Lecture 6

Then a theorem indicating solution accuracy is:

Theorem:

If the initial value problem for a linear PDE (for which the initial value

problem is well-posed), is approximated by a stable one-step finite

difference scheme which is rth order method accurate in space and sth

order in time (with r

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Approach 2) Compare Solution and

Numerical Solution at Nodes

It is trickier to perform a pointwise evaluation of thedifference between the numerical solution and the

exact solution.

The primary difficulty is that solutions in L2 are

equivalent if they only differ on a set of measure

zero.

Since the set of data points is a set of measure

zero the evaluation of an L2 solution at the points is

not well defined.

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cont

We rely on the following approximation result

Theorem 1.3.4 GKO:

Let u be a periodic function and assume that its Fourier

coefficients satisfy:

Then:

where the norm is the sup norm.

with 0, 1 mC

u m

1m

Mu I u CCdx

The assumption on the coefficients implies at least the mth order Fourier derivative exists.

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cont (sketch of convergence)

With this estimate in hand:

We consider:

1m

Mu I u CCdx

,

,

,

,

, ,

,

n

m n m dx

n

n mM n M n dx

n n n

n M n M n m m mdx

n n

m m

dx dx

u x t u

u t u

u t I u t I u t v

I u t I

v u

u t v v

Bound by approximation estimate

(assumes regularity of solution)

Compare interpolant of solution

and numerical solution started

with M term series truncation of

solution: nm

v

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Sketch cont

, ,,

,

m n M m n

n

M m n m dx

n n

m m dx

n

x

n

m m dx

dIu x t I u x t vu x t

u x t u

v u

i. Bounded by approximation result

ii. use well posedness to bound in

terms of the initial data

Bounded by approach 1 Bounded by

stability of method

and accuracy of

initial condition

Notice we use: regularity of the solution

well posedness of the initial value problem

comparison of interpolated numerical solution with solution

stability of method

accuracy of initial condition

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cont (final result)

We are left with the final accuracy estimate theorem

Theorem:

If the initial value problem for a PDE for which the initial value

problem is well posed, is approximated by a stable one-step

finite difference scheme that is rth order in space and sth order

in time with r

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Summary of Convergence Test for

Finite Difference Schemes

i. Is the PDE well posed (if in doubt look it up) ?

ii. Is the finite-difference method stable ?

i. use the method of lines

ii. a standard time-stepping method has a known region of

absolute stability bound for dt*maximum eigenvalue of

the spatial operator

iii. Is the finite-difference method consistent

i. use a Taylor series to estimate the local truncation in both

time and space

iv. what is the method order of accuracy ?

i. beware the case of low regularity initial data unbounded

remainder terms from Taylor series analysis

Finally, if the method order is p then the error analysis gives order p

in the solution (assuming the solution has p bounded Fourier derivatives)

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Boundary Conditions

We are now faced with the inevitable discussion ofhow to apply boundary conditions for a non-

periodic domain.

The advection equation only requires inflow data at

the node

nt

3Mx 2Mx 1xx

1Mx 0x

advection directionM

x

Mx

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Example Right-Difference

The obvious choice is to set the last node to be dxaway from the inflow boundary

1

inflow

0,1,..., 1m m mm

M

du u uc u c m M dt dx

u t u t

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System

An example system for 10 data points this timelooks like:

0 0 0

1 1 1

2 2 2

9 9 9 in

1 1

1 1

1 1

1 1

1 1

1 1

1 11 1

1 1

1

u u u

u u u

u u u

d d c cc

dt dx dx dx

u u u u

flow

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Interim (summation by parts) Result

First we define a discrete inner-product:

and associated norm:

Lemma (summation by parts formula):

This looks very much like an integration by parts formula, but

in this case with a discrete inner-product

,,

m s

r sr sm r

u v dx u v

2

, ,,

r s r su u u

1 1, , ,, , , s s r rr s r s r su v u v dx u v u v u v

E M th d f S i di t Diff A i ti

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Energy Method for Semi-discrete Difference Approximation

of the Upwind Finite Difference Method

We assume exact treatment of the time variableand 1st order upwind in the space derivative.

The left hand side represents the time rate change

of a numerical energy.

2

0, 1 0, 1

0, 1 0, 1

0, 1 0, 1

0 00, 1

,

, ,

, ,,

M M

M M

M M

M MM

d du u u

dt dt

du duu u

dt dt

c u u c u udxc u u c u u u u

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cont

The right hand side indicates that the scheme is

dissipative and the only source is from the inflow

boundary condition.

i.e. the total energy can only increase by input fromthe boundary condition

2

0 00, 1 0, 1

2

, M MM M

M

d u cdx u u c u u u udt

c u

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cont

Thus:

we are certainly in good shape (see GKO p448 forgeneralization to discrete in time and space).

The big but for this method is that it is first order in space

A method of lines analysis reveals that all the eigenvalues ofthe homogeneous operator arec/dx and then we have torely on further results on the impact of lower order terms (inthis case the boundary condition contribution) on the

stability of finite difference schemes

2 2

0, 1

2 2 2

inflow0, 1 0, 10

0

MM

t

M M

d u c udt

u t u c u d

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2nd Order Central + Boundary Conditions

Consider the following semi-discrete scheme:

We need special treatments for the two end points as the

stencil extends beyond the end of the data.

At the zero node we use the first order upwind condition:

At the Mth node we supply the inflow data as before.

0 for 1,2,..., 1

0

mm

m m

duc u m N

dt

u f x

0 1 00

du u uc u c

dt dx

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System

0 0 0

1 1 1

2 2 2

9 9 9

2 2

1 0 1

1 0 1

1 0 1

1 0 1

1 0 12

1 0 1

1 0 1

1 0 1

1 0

u u u

u u u

u u u

d d cc

dt dx dx

u u u

inflow

2

c

dx

u

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Interim (summation by parts) Result 2

We define a modified discrete inner-product:

and associated norm:

Lemma (summation by parts formula with central differences):

This looks very much like an integration by parts formula, but

in this case with a discrete inner-product

0 0 1, 1, : ,2 M Mdx Mdx

u v u v u v u v

2

,dx dxu u u

0 0 1 1 1 1, ,1

, , 2s s s s r r r r r s r su v u v u v u v u v u v

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Corollary

Setting v=u

This looks rather like a discrete analog of thedivergence theorem (or integration by parts).

The end point evaluation is now approximated byan average of the end point and neighbor.

0 0 1 1 1 1, ,

0 0 1 1 1 1, ,

1, ,

2

1, ,

2

s s s s r r r rr s r s

s s s s r r r rr s r s

u v u v u v u v u v u v

u u u v u u u u u u u u

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Example 2: Energy Method

The scheme:

has an energy equation (in the tailored norm):

0

inflow

0 1 0

0

for 1,2,..., 1

0

mm

M

m m

du c u m N dt

u u

du u u

c u cdt dx

u f x

22 2

0Mdx

du c u c u

dt

See GKO p452 for details

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Higher Order + Boundary Conditions

It is possible to apply modification to the higher order

central differences but it gets quite complicated.

We will reserve higher order treatment of boundaryconditions for finite-element and discontinuousGalerkin where it is more straightforward toaccommodate boundary conditions (i.e. I havechickened out).

For the interested, see GKO p474-484 for some nontrivial manipulations to the difference operators toaccommodate the boundary conditions in a stablemanner.

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Class Discussion

i. Pros and cons of finite difference methodsi. ease of implementation in simple geometries

ii. locality of derivatives

iii. cheap!!!!!!!!!!

iv. time stepping condition not generally artificially costly

v. difficulty of implementing boundary conditionsvi. technical difficulties in analysis

vii. spurious modes

ii. Boundary conditionsi. using extra points at boundary

ii. maintaining stabilityiii. one sided stencil interpolation

iii. Geometryi. stair stepping in 2D,3D

ii. embedded methods

N L

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Next Lecture

Higher order PDEs 2D and 3D domains

i.e. wrap up of finite-difference introduction.

Preparation for finite-volume methods.

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CAAM452Lecture8(1)