1.4 Convergence and Consistency
Q: Is as
Def: A one-step finite difference scheme approximating a partial differential equation is a convergent scheme if
For any solution to the partial differential equation , and solutions to the finite difference scheme, , such that converges to as mh converges to x, then converges to as (nk,mh) converges to (t,x) as h,k converge to 0.
Def: Given a partial differential equation
, and a finite difference scheme,
, we say the finite difference
scheme is consistent with the partial
differential equation if for any smooth function
as
the convergence being pointwise convergence at each grid point.
Ex: The forward-time forward-space Scheme
Let So
where
by Taylor series we have
So
Thus
as
This scheme is consistent
Def: If F and G are function of some parameter , we write
F=O(G) as
if
,
for some constant K and all sufficiently small.
F=o(G) as
if F/G converges to zero as tends to zero.
Examples:
F= means F goes to zero as fast/slow as
Ex: The Lax-Friedrichs Scheme
by Taylor series
So as and
Ex: with
We have shown the scheme is consistency.
Now , let
and
but for
Therefore
1.5 Stability
If so has to be bounded is some sense.
Def: A FDS for a first-order equation is stable in a stability region
If an integer J s.t. T>0 s.t
If we let
Then we have
or
i.e , The norm of the solution at any time , is limit in the amount of growth that can occur. Note that J might be 0
A stability region is any bounded nonempty region of the first quadrant of that has the origin as an accumulation point.
Example:
We show this is stable if
Proof`:
How about inhomogeneous problem?
In Chapter 9:
A scheme is stable for the if it is stable
For the equation .
The Lax-Richtmyer Equivalence Theorem
The importance of the concepts of consistency and stability is seen in the Lax-Richtmyer equivalence theorem, which is the fundamental theorem in the theory of finite difference schemes for initial value problem.
Theorem: A consistent finite difference scheme for a pde for which the initial value problem is well-posed is convergent if and only if it is stable.
Def: The initial value problem for the first-order partial differential equation Pu =0 is well-posed if for any time T>=0, there is a constant such that any solution satisfies
for
Consistency Convergence
stability
easy, concepts hard analysis
Remark: Non-stable scheme can not be convergent.
1.6 The Courant-Friedrichs-Lewy Condition
The CFL Condition
Example 1.5.1
is stable if
i.e. the Lax-Friedrichs scheme is stable
if , where
Claim: is necessary condition for stability
Theorem: For an explicit scheme for the hyperbolic equation (1.1.1) of the form
with
a necessary condition for stability is the
Courant-Friedrichs-Lewy(CFL) condition
pf: Suppose ,
Consider , it depends on either or
but depends on x only for
since depends on for
and
so
so it is not stable
Theorem: There are no explicit unconditionally stable, consistent finite difference schemes for hyperbolic systems of pdes.
Two implicit schemes which are unconditionally stable
backward-time center-space scheme
This will be showed in Section 2.2
backward-time backward-space scheme
if a>0