Transcript Document

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: (Lax’s 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.
Recall:
Test problem:
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.
Parceval’s 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
j–1/2
j–1
j+1/2
j
j+1
unit volume
Define a flux at the interface j–1/2, j+1/2,
explicit Euler scheme in time as
, then discretize PDE using
However, there is no grid point (no data!) at j-1/2, j+1/2…
 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
j–1/2
And the third term as
j–2
j–1
j+1/2
j
j+1
j–2
k– scheme (continued)
Deriving a FD scheme explicitly for wnj , one finds that the coefficients of wnj
Are effectively proportional to k – 2 m|n| . Hence one parameter may be eliminated.
A choice
results in the form of k – scheme derived by Van Leer.
k – scheme becomes
k = 1/3 Quickest scheme
Leonard (1979)
k = 1/2 Quick scheme
k = 0 Fromm scheme (optimal)
Different from the Van Leer’s choice
m= k = 1
Lax-Wendroff
k – 2mn = –1 Warming & Beam
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 nonincreasing (decreasing ) initial data
the numerical solution
is non-increasing (decreasing).
Godunov’s thorem
For the uniform grid
and the constant time step
the (explicit or implicit) one-step scheme, in which
at the
(n+1)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: (Godunov’s 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.
Godunov’s thorem (continued)
For the linear s-step multi-step scheme, the same Godunov’s 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)
• Godunov’s 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 > 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
• Let the flux limiter
satisfies
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 B=0 line for
are allowed.
Lax-Wendroff: B = 1
Warming Beam: B = r
If
1
1
0
(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.
Lax-Wendroff: B = 1
2
1
Warming Beam: B = r
1
0
Sweby (1985) showed that the
admissible limiter regions for the
2nd order TVD scheme are those
bounded by these two limiters.
Schemes that is 2nd order in the
smooth flow region, and do not
oscillate at the discontinuity is
called high resolution scheme.
Minmod limiter
Superbee limiter
2
TVD
1
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.