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, [346Jump To The Next Citation Point]. 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.


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