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7. Introduction to the numerical integration of PDE.

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Title: 7. Introduction to the numerical integration of PDE.

1
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.

2
Accuracy requirements

Usually t is more restricted by stability than by
accuracy.
Notation for the discretization of
3
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)
4
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 .
5
lp norm and l1 norm are defined by
Theorem (Laxs equivalence theorem) Convergence
) Zero (or Absolute) - stability. Zero (or
Absolute) - stability and Consistency )
Convergence.
6
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
7
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.)
8
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
9
The von Neumann condition for zero stability
The von Neumann condition for absolute stability
10
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.
11
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).
12
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
13
(1) Explicit Euler
(2) Lax
(3) Lax-Wendroff
(4) 1st order upwind
14
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
15
• 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.
16
Monotonicity preservation of a linear advection
equation
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).
17
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.

18
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,
19
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
20
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.
21
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.)
22
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

23
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
24
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
25
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.