Loading...

PPT – 7. Introduction to the numerical integration of PDE. PowerPoint presentation | free to download - id: 67ed76-ZjZkY

The Adobe Flash plugin is needed to view this content

7. Introduction to the numerical integration of

PDE.

As an example, we consider the following PDE

with one variable

- Finite difference method is one of numerical

method for the PDE.

Accuracy requirements

Usually t is more restricted by stability than by

accuracy.

Notation for the discretization of

Summary of the key concept on numerical method

for PDE.

- Local truncation error The amount that

the exact solution of PDE fails to - satisfy the the finite difference equation.

ex.) One-step method.

- Global discretization error

Definitions (Consistency)

Definitions (Convergence)

Definitions (Zero-Stability - a stability

criteria with h ! 0 )

Definitions (Absolute Stability - a stability

criteria with a fixed h)

Remark Purpose of stability analysis is to

determine t0(h) which guarantee that the

perturbation does not glow. This is the case if

Note that for the case of Zero-stability, the

dimension (size) of the matrix C0(h) increase as

n ! 1 .

lp norm and l1 norm are defined by

Theorem (Laxs equivalence theorem) Convergence

) Zero (or Absolute) - stability. Zero (or

Absolute) - stability and Consistency )

Convergence.

Some explicit integration method for a linear

wave equation.

- Some basic schemes for are presented.

(1) FTCS (Forward in Time and Central difference

Scheme).

(2) Lax (- Friedrich) scheme.

(3) Leap-Flog scheme

(4) Lax-Wendroff scheme

(5) 1st order upwind scheme

Some explicit integration method for a linear

wave equation continued.

- Lax, Lax-Wendroff, 1st order upwind schemes

can be understood as - FTCS scheme .Diffusion term.

(1) Explicit Euler scheme (FTCS Forward in

Time and Central difference).

(2) Lax (- Friedrich) scheme.

(3) Lax-Wendroff scheme.

(4) 1st order upwind scheme

(weakest diffusion)

(Can be used also for negative c.)

von Neumann stability analysis.

- A method to analyze the stability of numerical

scheme for linear PDE - (assuming equally spaced grid points and

periodic boundary condition). - Consider that the finite difference equation

has the following solution.

- Then the perturbation

can be also written

Substituting the Fourier transform above, we have

Amplification factor g(q) is defined by

The von Neumann condition for zero stability

The von Neumann condition for absolute stability

Another derivation of von Neumann stability

condition.

Then we apply absolute stability condition for

the ODE.

Test problem

Recall

Characteristic polynomial Q

Definition The region of absolute stability for

a one-step method is the set

Therefore for the PDE, the region of absolute

stability is the set

Note

l2 norm. Parcevals theorem.

Characteristic polynomials for a particular FD

scheme.

A substitution of

to a FD equation, and the mode

decomposition results in a equation for each

Fourier component . Then, substituting

we have characteristic polynomials.

A solution to the polynomial becomes an

amplification factor for each mode q.

If the exact values

are known, conveniently shows

amplification rate and phase error of each

mode.

Note q 0, corresponds to low frequency (long

wavelength). q p, corresponds to high

frequency (short wavelength).

Finite volume discretization

Another concept for deriving finite difference

approximation suitable for the conservation law

PDEs of the conservative form

j1/2

j1/2

j1

j

j1

unit volume

Define a flux at the interface j1/2, j1/2,

, then discretize PDE using explicit Euler

scheme in time as

However, there is no grid point (no data!) at

j-1/2, j1/2 ? Use

to evaluate

One can rewrite explicit Euler, Lax,

Lax-Wendroff, 1st order upwind using

(1) Explicit Euler

(1) Explicit Euler

(2) Lax

(3) Lax-Wendroff

(4) 1st order upwind

k scheme A parametrization of representative

linear schemes. (Van Leer)

For a linear PDE , write a Taylor expansion in

time

Approximate the second term in RHS as

j1/2

j1/2

And the third term as

j2

j2

j1

j

j1

- scheme (continued)

Deriving a FD scheme explicitly for wnj , one

finds that the coefficients of wnj Are

effectively proportional to k 2 mn . Hence

one parameter may be eliminated. A choice

results in the form of k scheme derived by Van

Leer.

k scheme becomes

- 1/3 Quickest scheme
- 1/2 Quick scheme
- 0 Fromm scheme (optimal)

Leonard (1979)

- k 1 Lax-Wendroff
- k 2mn 1 Warming Beam

Different from the Van Leers choice

Method of lines (yet another idea for

discretization.)

In the k scheme (and the linear scheme we have

seen) a dependence on the time step t is

included in the Courant number n. To avoid this,

one discretizes PDE along spatial direction

first as For the FD operator Lh , choose e.g. k

scheme, then apply ODE integration scheme such

as RK4.

Monotonicity preservation of a linear advection

equation

A linear advection equation preserves

monotonicity i.e. if f(0,x) monotonic ) f(t,x)

monotonic, since its general solution is .

Consider a finite difference scheme that

generates numerical approximation to

. data at the time step n.

Definition (Monotonicity preserving scheme) A

numerical scheme is called monotonicity

preserving if for every non-increasing

(decreasing ) initial data the numerical

solution is non-increasing

(decreasing).

Godunovs thorem

For the uniform grid and the constant

time step the (explicit or implicit) one-step

scheme, in which at the (n1)th step is

uniquely determined from at the nth

step, is written

Theorem (Godunov Monotonicity preservation) The

above one-step scheme is monotonicity preserving

if and only if

Theorem (Godunovs order barrier theorem) Linear

one-step second-order accurate numerical schemes

for the convection equation cannot be

monotonicity preserving, unless

- Remarks
- If the numerical scheme keeps the

monotonicity, a numerical solution do - not shows (unphysical) oscillations (such as

at the discontinuity). - In these theorems, the stencils cm for the

one-step FD formula are assumed - to be the same at all grid points (Linear

scheme). - Practically, one can not have the 2nd order

linear one-step scheme.

Godunovs thorem (continued)

For the linear s-step multi-step scheme, the same

Godunovs theorems holds.

cf.) Local truncation error of the linear one

step scheme,

Two 2nd order schemes Lax-Wendroff and Warming

Beam schemes

From the local truncatoin error formula,

the 2nd order scheme needs to satisfy

Choice of 3 grid points j 1, j, j 1, (m 1,

0, 1) results in Lax-Wendroff.

( Explicit Euler Diffusion term centered at j.)

Choice of 3 grid points j 2, j 1, j, (m 2,

1, 0) results in Warming Beam.

( 1st order upwind Diffusion term centered at

j-1.)

Writing these in the flux form

Lax-Wendroff

Warming Beam

Total variation diminishing (TVD) property.

- Total variation of a function TV(f) is

defined by

Definition (TVD). If TV(f) does not increase

in time, f(t,x) is called total variation

diminishing or TVD.

- For f(t,x) a solution to ,

, we have

which is independent of t, hence f(t,x) is TVD.

This motivates to derive a numerical scheme whose

total variation of a solution

does not increase in time step,

Definition A numerical scheme with this property

is called TVD scheme.

Theorem (TVD property)

The scheme is TVD if and only if

Corollary TVD scheme is monotonicity

preserving.

Monotonicity preserving scheme with flux limiter

function. (Flux limted schemes)

- Godunovs theorem does not allow the 2nd order

linear one-step scheme. - Conditions to be satisfied by the 1st order
- monotonicity preserving scheme are
- Considering that the number of grid
- points for the 1st order scheme are 2 points,
- resulting scheme is the 1st-order upwind.

Lax-Wendroff scheme is understood as modifying

the flux of 1st order upwind.

1st order upwind ( c gt 0 )

Lax-Wendroff

Consider a non-linear scheme that modify the flux

with a limiter function

(The value of differs at each cell boundary.)

Condition for the flux with a limiter function to

be monotonicity preserving.

- Derive sufficient condition for the scheme
- with the flux
- to be the monotonicity preserving.

Substituting the flux in the scheme,

- Sufficient condition for the scheme to be

monotonic is

This is satisfied if the flux limiter function

satisfies

- Let the flux limiter to be a function of

the slope ratio

Sufficient region for to have

monotonicity preserving scheme.

2

White region in the right panel for and B0 line

for are allowed.

1

Lax-Wendroff B 1

1

Warming Beam B r

0

If (i.e. the flow is not monotonic at

rj ) ) ) 1st order upwind.

If , many choices. It is desirable

to have 2nd order scheme for a smooth flow

around rj 1.

1st order upwind

Lax-Wendroff

Warming Beam

Minmod limiter and Superbee limiter and high

resolution scheme.

is called the flux limiter function, or the slope

limiter function.

Minmod and Superbee are two representative

limiters.

2

Lax-Wendroff B 1

1

Warming Beam B r

1

0

Superbee limiter

Minmod limiter

Sweby (1985) showed that the admissible limiter

regions for the 2nd order TVD scheme are those

bounded by these two limiters.

2

TVD

1

Schemes that is 2nd order in the smooth flow

region, and do not oscillate at the discontinuity

is called high resolution scheme.

1

0

Next steps.

- Numerical schemes for the conservation laws

(non-linear PDE). - Including understand characteristics
- introduction of weak solutions and

shocks. - introduction of monotonicity and TVD

property. - conservative form of FD schemes.
- application of various numerical

schemes - (linear schemes, Godunov scheme,
- high resolution schemes (MUSCL),

artificial viscosity etc.)

- Numerical schemes for the system equations

ex) the Euler system - Including characteristics, shocks and

Rankine-Hugoniot conditions. - application of various numerical

schemes - approximate Riemann solver, (Godunov

scheme, Roe scheme) - High resolution schemes (MUSCL)

- Discretization in higher dimension and general

domain.