8.4 Stability

If a simple symmetric hyperbolic system with constant coefficients in one dimension of the form
ut = Aux (8.36 )
with T A = A symmetric, is discretized in space by approximating the space derivative with an operator satisfying SBP, a semi-discrete energy estimate can be derived, modulo boundary conditions (discussed in Section 10). For this, we define
Ed (t) := ⟨u,u⟩Σ. (8.37 )
Taking a time derivative, using the symmetry of A, the fact that D and A commute because the latter has constant coefficients, and the SBP property,
dEd- d- -d [ T ]b dt = ⟨dtu,u ⟩Σ + ⟨u,dtu ⟩Σ = ⟨DAu, u⟩Σ + ⟨Au, Du ⟩Σ = (Au ) u a. (8.38 )
Therefore, modulo the boundary terms, an energy estimate and semi-discrete stability follow. As usual, the addition of lower-order undifferentiated terms to the right-hand side of Eq. (8.36View Equation) still gives an energy estimate at the semi-discrete level, modulo boundary terms.

When considering variable coefficients, Eq. (8.38View Equation) becomes

dE [ ] ---d = (Au )Tu b− ⟨u,[D,A ]u⟩Σ (8.39 ) dt a
and the commutator between A and D needs to be uniformly bounded for all resolutions in order to obtain an energy estimate.

Estimates for the term involving the commutator [D, A ] have been given in [317Jump To The Next Citation Point, 319Jump To The Next Citation Point, 318Jump To The Next Citation Point, 407Jump To The Next Citation Point]. In order to discuss them, we first notice that the SBP property of D with respect to Σ and the symmetry of A imply that the operator B := [D, A ] is symmetric with respect to Σ,

⟨u,[D, A]v⟩Σ = ⟨[D, A ]u, v⟩Σ (8.40 )
for all grid functions u and v. Therefore, its norm is equal to its spectral radius and we have the estimate
⟨u,[D, A ]u ⟩Σ ≤ ρ([D, A])∥u∥2Σ (8.41 )
for all grid functions u. Hence, the problem is reduced to finding an upper bound for the spectral radius of [D, A ], which is independent of the scalar product Σ.

Next, in order to find such an upper bound, we write the FD operator as

N -1--∑ (Du )j = Δx djkuk, j = 0,1,...N, (8.42 ) k=0
where the djk’s are the coefficient of a banded matrix; that is, there exists b > 0, which is independent of N such that d = 0 jk for |k − j| > b. Then, we have
∑N A (xk) − A(xj) Buj = [D, A ]uj = djk-------------- uk, j = 0,1, ...N, (8.43 ) k=0 Δx
from which it follows, under the assumption that A is continuously differentiable and that its derivative is bounded, N∑ |Buj | ≤ |Ax |∞ |k − j||djk||uk| k=0, j = 0, 1,...N, where |Ax |∞ := sup |Ax (x)| a≤x≤b. Now we can easily estimate the spectral radius of B, based on the simple observation that
ρ(B ) ≤ ∥B ∥, ∥B∥ := sup ∥Bu-∥-, (8.44 ) u⁄=0 ∥u ∥
for any norm ∥ ⋅ ∥ on the space of grid functions u. Choosing the 1-norm ∑N ∥u∥ := |uj| j=0 we find
( ) N∑ ∑N ∑N ∥Bu ∥ = |Buj | ≤ |Ax |∞ |k − j||djk||uk| ≤ |Ax|∞ max |k − j||djk| ∥u ∥, (8.45 ) j=0 j,k=0 k=0,...,N j=0
from which it follows that
∑N ρ([D, A ]) ≤ C1 |Ax |∞, C1 := max |k − j||djk|. (8.46 ) k=0,...,N j=0
The important point to notice here is that for each fixed k, the sum in the expression for C1 involves at most 2b + 1 non-vanishing terms, since djk = 0 for |k − j| > b. For the SBP operators and scalar products used in practice, namely those for which the latter has the structure given by Eq. (8.29View Equation), C 1 can be bounded by a constant, which is independent of resolution. Since the spectral radius of B is equal to that of its transposed, we may interchange the roles of j and k in the definition of the constant C1, and with these observations we arrive at the following result:

Lemma 7 (Discrete commutator estimate). Consider a FD operator D of the form (8.42View Equation), which satisfies the SBP property with respect to a scalar product Σ, and let A = AT be symmetric. Then, the following commutator estimate holds:

|⟨u,[D, A]u⟩ | ≤ C |A | ∥u∥2 (8.47 ) Σ x ∞ Σ
for all grid functions u and resolutions N, where C := min {C1,C2 } with
∑N N∑ C1 := max |k − j||djk|, C2 := max |k − j||djk|. (8.48 ) k=0,...,N j=0 j=0,...,N k=0

Remarks


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