Unformatted text preview: n (3.2.4a) to the
above equation to obtain kU n Since jMk j
nd J ;1
2 = J X jMk j2n jA0 j2 :
k2
k
k =0 1 for all k when 0 < kUnk2 J
2 1, we can use Parseval's relation once more to J ;1
X jA0 j2 = kU0k2:
k
2 k =0
2 stability of (2.2.4) when the Courant number is a constant
This establishes the L 2 (0 1].
Example 3.2.2. Let us, at long last, use von Neumann's method to examine the
L2 stability of the forward timecentered space scheme (2.2.5) for the kinematic wave 16 Basic Theoretical Concepts n
equation (2.2.1). Necessarily, we assume that j is a constant and the initial data is
periodic in j with period J so that (2.2.5) becomes Ujn+1 = Ujn ; 2 (Ujn+1 ; Ujn;1):
Expanding Ujn in a discrete Fourier series (3.2.1a), we nd
J ;1
X
k =0 An+1 ; An + 2 (e2 ik=J ; e;2 ik=J )An]!jk = 0:
k
k
k Once again, the orthogonality relation (3.2.2) may be used to infer that An = (Mk )nA0
k
k
where Mk = 1 ; 2 (e2 ik=J ; e;2 ik=J ) = 1 ; i sin 2 k : The magnitude of the ampli cation factor is J jMk j2 = 1 + 2 sin2 2J k :
There is no possibility of restricting so that jMk j 1 for all k thus, most Fourier modes
will grow. As bad as this appears, (2.2.5) is still stable according to either (3.1.14a) or
(3.1.14b). To demonstrate this, observe that jMk j2 1 + 2:
The following inequality is useful in many situations, so we'll record it as a lemma. Lemma 3.2.1. For all real z
1+z ez : Proof. Using Taylor's formula for the exponential function ez = 1 + z + 1 e z2 1 + z:
2 (3.2.5) 3.2. Fourier Stability Analysis 17 In the present case, (3.2.5) can be used to bound the ampli cation factor as jMk j2 e :
2 Taking the nth power and using the de nition of the Courant number (2.2.3) jMk j2n ena 2 t2 = x2 = e(n t)(a2 t= x2 ): As usual, introduce T so that n t T . Also, let us agree to select meshes having
temporal and spatial steps satisfying a2 t= x2 = , where is a constant. Then jMk j2n eT = C 2 :
Using Parseval's relation (3.2.4a) and following the steps used in Example 3.2.1, we nd kU n J ;1
2 = J X jMk j2n jA0 j2
k2
k
k =0 C 2kU0 k2:
2 We have found a constant C satisfying kUnk2 C kU0k2
hence, (2.2.5) is stable in L2. The constant C is greater than unity, so that most Fourier
modes will grow. In fact, if T and are not small enough, initial disturbances will grow
to such an extent that they dominate the computation. Additionally, we are forced to
restrict the time step t
x2 =a2, while the directional methods (2.2.2b, 4) only
require t
x=a. This leads us to conclude that, while technically stable, the forward
timecentered space scheme (2.2.5) is not a practical method to apply to the kinematic
wave equation (2.2.1).
When applying von Neumann's method we will always nd a solution of the form Ujn = J ;1
X
k =0 (Mk )nA0 !jk :
k In Example 3.2.1, we found that jMk j 1 so that initial perturbations did not grow and
the forward timebackward space scheme was stable in L2. In Example 3.2.2, jMk j > 1
and perturbations grew, but were bounded for nite times T . While the (forward timecentered space) scheme was stable, we saw that it was impractical and, thus, may ask 18 Basic Theoretical Concepts whether or not we should keep jMk j 1 as a practical matter. The answer to this
question depends on the behavior of the problem under investigation. If the solution
of the partial di erential equation is unstable and, hence, growing in time, then some
growth of initial perturbations can and should be tolerated. However, if the solution of
the partial di erential equation does not grow, then either jMk j must be bounded by
unity or we must be willing to compute for su ciently small times so that the bound C
on the growth of kUnk remains small. There are only a few instances where the latter
option will be desirable thus, we should typically maintain jMk j 1 whenever solutions
of the partial di erential equation under study do not grow.
In those cases where jMk j may exceed unity, we may ask the amount by which it may
do so. This leads to the von Neumann condition. De nition 3.2.1. A nite di erence scheme satis es the von Neumann condition if
there exists a positive constant c that is independent of t, x, and k which satis es jMk j 1 + c t 8t t x x: (3.2.6) Finite di erence schemes that satisfy the von Neumann conditio...
View
Full Document
 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Fourier Series, Partial differential equation, Hilbert space, di erence, nite di erence

Click to edit the document details