Notation and Conventions

2 Notation and Conventions

Throughout this article, we use the following notation and conventions. For a complex vector $u ∈ ℂm$ , we denote by $u∗$ its transposed, complex conjugate, such that $u ⋅ v := u∗v$ is the standard scalar product for two vectors $u, v ∈ ℂm$ . The corresponding norm is defined by $|u| := √u-∗u$ . The norm of a complex, $m × k$ matrix $A$ is

|Au | |A| := sup -----. u∈ℂk∖{0} |u|

The transposed, complex conjugate of $A$ is denoted by $A∗$ , such that $v ⋅ (Au ) = (A ∗v) ⋅ u$ for all $k u ∈ ℂ$ and $m v ∈ ℂ$ . For two Hermitian $m × m$ matrices $∗ A = A$ and $∗ B = B$ , the inequality $A ≤ B$ means $u ⋅ Au ≤ u ⋅ Bu$ for all $m u ∈ ℂ$ . The identity matrix is denoted by $I$ .

The spectrum of a complex, $m × m$ matrix $A$ is the set of all eigenvalues of $A$ ,

σ (A) := {λ ∈ ℂ : λI − A is not invertible},

which is real for Hermitian matrices. The spectral radius of $A$ is defined as

ρ(A ) := max {|λ | : λ ∈ σ (A)}.

Then, the matrix norm $|B|$ of a complex $m × k$ matrix $B$ can also be computed as $∘ -------- |B | = ρ(B∗B )$ .

Next, we denote by $2 L (U )$ the class of measurable functions $n m f : U ⊂ ℝ → ℂ$ on the open subset $U$ of $ℝn$ , which are square-integrable. Two functions $f,g ∈ L2 (U )$ , which differ from each other only by a set of measure zero, are identified. The scalar product on $L2 (U)$ is defined as

∫ ⟨f,g⟩ := f(x)∗g(x)dnx, f,g ∈ L2 (U), U

and the corresponding norm is $∘ ------ ∥f∥ := ⟨f, f⟩$ . According to the Cauchy–Schwarz inequality we have

⟨f,g⟩ ≤ ∥f ∥∥g∥, f,g ∈ L2(U ).

The Fourier transform of a function $f$ , belonging to the class $C ∞(ℝn ) 0$ of infinitely-differentiable functions with compact support, is defined as

∫ 1 −ik⋅x n n fˆ(k) := (2π)n∕2 e f(x )d x, k ∈ ℝ .

According to Parseval’s identities, $⟨ˆf,ˆg⟩ = ⟨f,g ⟩$ for all $∞ n f,g ∈ C 0 (ℝ )$ , and the map $∞ n 2 n C0 (ℝ ) → L (ℝ )$ , $f ↦→ fˆ$ can be extended to a linear, unitary map $ℱ : L2(ℝn ) → L2 (ℝn)$ called the Fourier–Plancharel operator; see, for example, [346 ]. Its inverse is given by $ℱ −1(f)(x) = fˆ(− x )$ for $f ∈ L2(ℝn )$ and $x ∈ ℝn$ .

For a differentiable function $u$ , we denote by $ut$ , $ux$ , $uy$ , $uz$ its partial derivatives with respect to $t$ , $x$ , $y$ , $z$ .

Indices labeling gridpoints and number of basis functions range from $0$ to $N$ . Superscripts and subscripts are used to denote the numerical solution at some discrete timestep and gridpoint, as in

vkj := v(tk,xj).

We use boldface fonts for gridfunctions, as in

k N v := {v(tk,xj)}j=0.