Some examples of kinetic schemes whose diffusion limit is Il’in’s exponential-fitting

Abstract

This paper is concerned with diffusive approximations of some numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov–Fokker–Planck plasma modeling. The well-balanced method applied to such kinetic equations leads to time-marching schemes involving a “scattering S-matrix”, itself derived from a normal modes decomposition of the stationary solution. One common feature these models share is the type of diffusive approximation: their macroscopic densities solve drift-diffusion systems, for which a distinguished numerical scheme is Il’in/Scharfetter–Gummel’s “exponential fitting” discretization. We prove that these well-balanced schemes relax, within a parabolic rescaling, towards such type of discretization by means of an appropriate decomposition of the S-matrix, hence are asymptotic preserving.

Appendices

A Haar property, Chebyshev T-systems and Markov systems

We first recall basic notions from standard (one-dimensional) approximation theory, following mostly [18, Chapter 3].

Definition A.1

Let $n \in {\mathbb {N}}$ and $F_n=(f_1, f_2, \ldots , f_n)$ be a family of functions, continuous on an interval $I \subset {\mathbb {R}}$: it is endowed with the Haar property if, for any strictly increasing family $X=(x_1, x_2, \ldots , x_n) \in I^n$, the family of n vectors $(f_1(X), f_2(X), \ldots , f_n(X))$ is linearly independent. Equivalently, the determinant never vanishes: $\forall \,(x_1, x_2, \ldots , x_n) \in I^n,\ x_1<x_2<\cdots <x_n$,

$$\begin{aligned} \left| \begin{array}{cccc} f_1(x_1) &{} f_2(x_1)&{} \cdots &{} f_n(x_1) \\ f_1(x_2) &{} f_2(x_2)&{} \cdots &{} f_n(x_2) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ f_1(x_n) &{} f_2(x_n)&{} \cdots &{} f_n(x_n) \\ \end{array}\right| \not = 0. \end{aligned}$$

(A.1)

Such a family $F_n$ constitutes a $\underline{\hbox {Chebyshev}~T\hbox {-system}}$ on the interval I.

The simplest example of T-system on $I={\mathbb {R}}$ is the monomials family, for which the determinant (A.1) is the well-known Vandermonde determinant.

Definition A.2

Let $F=(f_1, f_2, \ldots )$ be an infinite sequence of functions, continuous on an interval $I \subset {\mathbb {R}}$: it is said to be a Markov system if, for any $n \in {\mathbb {N}}$, the extracted finite family $F_n \subset F$ is a Chebyshev T-system.

A standard, yet important result is the following:

Proposition A.1

Let $n \in {\mathbb {N}}$ and $F_n=(f_1, f_2, \ldots f_n)$ be as in Definition A.1: it is a T-system on I if and only if any (real, and non trivial) linear combination,

$$\begin{aligned} \forall (a_1, a_2, \ldots , a_n) \in {\mathbb {R}}^n, \qquad I \ni x \mapsto \sum _{i=1}^n a_i\, f_i(x), \end{aligned}$$

(A.2)

admits at most $n-1$ real roots on I.

Remark A.1

Determinants of the type (A.1) are called “alternant determinants”, see [57, Chapter 4]. Moreover, the Haar property is closely related to “total positivity” of matrices, see e.g. [24]. Being the “Hadamard product” the component-wise product of two $n \times n$ matrices,

$$\begin{aligned} (A \circ B)_{1 \le i,j \le n}:= A_{i,j}\, B_{i,j}, \end{aligned}$$

Garloff and Wagner in [24, page 100], explain that the Haar property is not generally preserved by multiplying elements of two T-systems with each other. A first exception is given by two generalized Vandermonde matrices $G_n=(x_i^{\alpha _j})_{1 \le i,j, n}$ sharing either the set of points X or the exponents $\alpha _i$’s. A second one is given by non-negative exponential monomials: see Theorem D.1.

B. Some properties on Case’s eigenelements

In this Appendix, we establish some useful properties on the eigenfunctions defined in Proposition 3.1. First we show that the set of Case’s eigenfunctions is endowed with the Haar property (see Definition A.1).

Proposition B.1

Denote $\phi _\lambda (v)= \frac{1}{1-\lambda \, v}$ for $\lambda \ge 0$, then the following properties hold:

• (i) Let $0<\lambda _1<\cdots < \lambda _{K-1}$ and $0<v_1<\cdots <v_K$. We denote ${{\mathcal {V}}}=(v_1,\ldots ,v_K)^\top$. Then, the family $\{{\mathbf {1}}_{{\mathbb {R}}^K}, \phi _{\lambda _1}({{\mathcal {V}}}), \ldots , \phi _{\lambda _{K-1}}({{\mathcal {V}}}) \}$ is a basis of ${\mathbb {R}}^K$.

• (ii) The set $(\phi _{\lambda })_{\lambda \ge 0}$ is a Markov system on ${\mathbb {R}}^+_*$ in the sense of Definition A.2.

• (iii) There exists $\beta \in {\mathbb {R}}^K$ and $\gamma \in {{\mathcal {M}}}_{K-1\times K}({\mathbb {R}})$ such that

  $$\begin{aligned} \begin{pmatrix} \beta ^\top \\ \gamma \end{pmatrix} = \Big ({\mathbf {1}}_{{\mathbb {R}}^K} \quad \phi _{\lambda _1}({{\mathcal {V}}}) \quad \cdots \quad \phi _{\lambda _{K-1}}({{\mathcal {V}}})\Big )^{-1}. \end{aligned}$$

Proof

These properties are shown by studying convenient polynomials.

• (i) As the family contains K vectors, it suffices to show its linear independence: Assume

  $$\begin{aligned} \exists a_0, a_1, \ldots , a_K, \qquad a_0 {\mathbf {1}}_{{\mathbb {R}}^K} + \sum _{i=1}^{K-1} a_i \phi _{\lambda _i}({{\mathcal {V}}}) = {\mathbf {0}}_{{\mathbb {R}}^K}, \end{aligned}$$

  let us show that $a_0=a_1=\cdots =a_{k-1}=0$. Using the expression of $\phi _\lambda$ and multiplying,

  $$\begin{aligned} \forall k\in \{1,\ldots ,K\}, \qquad \left( a_0 + \sum _{i=1}^{K-1} \frac{a_i }{1-\lambda _i v_k}\right) \times \prod _{j=1}^{K-1} (1-\lambda _j v_k) = 0. \end{aligned}$$

  Thus, for any $k\in \{1,\ldots ,K\}$, the polynomial

  $$\begin{aligned} v \mapsto P(v) := a_0 \prod _{j=1}^{K-1} (1-\lambda _j v) + \sum _{i=1}^{K-1} a_i \prod _{j=1,j\ne i}^{K-1} (1-\lambda _j v) \end{aligned}$$

  has degree $K-1$, but K roots $\{v_1,\ldots ,v_K\}$, so that P is a null polynomial, i.e.

  $$\begin{aligned} \forall v\in {\mathbb {R}}, \qquad P(v) = 0. \end{aligned}$$

  Identifying the term of higher degree, we deduce that $a_0=0$. Then, taking $v=1/\lambda _i$, $i\in \{1,\ldots ,K-1\}$, we obtain $a_i=0$, for all $i\in \{1,\ldots ,K-1\}$.

• (ii) From (i), since the values of both $K \in {\mathbb {N}}$ and $\lambda$’s are arbitrary, the set $(\phi _\lambda (v))_{\lambda \ge 0}$ clearly constitutes a Markov system.

• (iii) The point (i) implies that the matrix $\Big ({\mathbf {1}}_{{\mathbb {R}}^K} \quad \phi _{\lambda _1}({{\mathcal {V}}}) \quad \cdots \quad \phi _{\lambda _{K-1}}({{\mathcal {V}}})\Big )$ is invertible. $\square$

For our next property, we consider two sets of positive numbers $0<\lambda _1<\cdots <\lambda _{K-1}$ and $0<\mu _1<\cdots <\mu _{K-1}$ with corresponding Case’s eigenfunctions $(\phi _\lambda )_\lambda$ and $(\phi _\mu )_\mu$. We denote $\gamma _1$, respectively $\gamma _2$, the corresponding matrices defined in Proposition B.1 (iii) for the set $(\lambda _i)_i$, respectively $(\mu _i)$. We introduce

$$\begin{aligned}&\zeta _1=\Big (\phi _{\lambda _1}({{\mathcal {V}}})-\phi _{\lambda _1}(-{{\mathcal {V}}}), \ldots , \phi _{\lambda _{K-1}}({{\mathcal {V}}})-\phi _{\lambda _{K-1}}(-{{\mathcal {V}}})\Big ) \\&\zeta _2=\Big (\phi _{\mu _1}({{\mathcal {V}}})-\phi _{\mu _1}(-{{\mathcal {V}}}), \ldots , \phi _{\mu _{K-1}}({{\mathcal {V}}})-\phi _{\mu _{K-1}}(-{{\mathcal {V}}})\Big ) \end{aligned}$$

Then, let us denote

$$\begin{aligned} {\mathcal {H}} = \begin{pmatrix} {\mathbf {I}}_K &{} \zeta _2\gamma _2 - {\mathbf {I}}_K \\ \zeta _1\gamma _1 - {\mathbf {I}}_K &{} {\mathbf {I}}_K \end{pmatrix}. \end{aligned}$$

The following Lemma shed light onto the kernel and the range of ${\mathcal {H}}$:

Lemma B.1

With the above notations, let us assume moreover that

$$\begin{aligned} \forall \, i, \quad \sum _{k=1}^K \omega _k v_k (\phi _{\lambda _i}(v_k)-\phi _{\lambda _i}(-v_k)) = \sum _{k=1}^K \omega _k v_k (\phi _{\mu _i}(v_k)-\phi _{\mu _i}(-v_k)) = 0. \end{aligned}$$

Then, the matrix ${\mathcal {H}}$ is such that:

• Ker$({\mathcal {H}})= \text{ Span }({\mathbf {1}}_{{\mathbb {R}}^{2K}})$,

• Im$({\mathcal {H}})= \Big \{Z= (Z_1\ Z_2)^\top ,\ Z_i\in {\mathbb {R}}^{K} \text{ such } \text{ that } \sum _{k=1}^K \omega _k ({Z_1}_k+{Z_2}_{k}) = 0\Big \}$.

Proof

• Pick $Y=(Y_1\ Y_2)^\top \in \text{ Ker }({\mathcal {H}})$, then

  $$\begin{aligned} Y_1-Y_2 = \zeta _1 \gamma _1 Y_1 = -\zeta _2 \gamma _2 Y_2. \end{aligned}$$

  Since, from Proposition B.1, the families $\{{\mathbf {1}}_{{\mathbb {R}}^K},\phi _{\lambda _1}({{\mathcal {V}}}),\ldots ,\phi _{\lambda _{K-1}}({{\mathcal {V}}})\}$ and $\{{\mathbf {1}}_{{\mathbb {R}}^K},\phi _{\mu _1}({{\mathcal {V}}}),\ldots ,\phi _{\mu _{K-1}}({{\mathcal {V}}})\}$ are basis of ${\mathbb {R}}^K$, we may write

  $$\begin{aligned} Y_1 = a_0 + \sum _{\ell =1}^{K-1} a_\ell \phi _{\lambda _\ell }({{\mathcal {V}}}), \qquad Y_2 = b_0 + \sum _{\ell =1}^{K-1} b_\ell \phi _{\mu _\ell }({{\mathcal {V}}}). \end{aligned}$$

  By definition of $\zeta _i$ and $\gamma _i$, $i=1,2$, we have

  $$\begin{aligned} \zeta _1\gamma _1 Y_1 = \sum _{\ell =1}^{K-1} a_\ell (\phi _{\lambda _\ell }({{\mathcal {V}}})-\phi _{\lambda _\ell }(-{{\mathcal {V}}})), \qquad \zeta _2\gamma _2 Y_2 = \sum _{\ell =1}^{K-1} b_\ell (\phi _{\mu _\ell }({{\mathcal {V}}})-\phi _{\mu _\ell }(-{{\mathcal {V}}})). \end{aligned}$$

  Thus from the equalities $Y_1=Y_2-\zeta _2\gamma _2 Y_2$ and $Y_2=Y_1-\zeta _1\gamma _1 Y_1$, we deduce

  $$\begin{aligned}&a_0-b_0+\sum _{\ell =1}^{K-1} \Big (a_\ell \phi _{\lambda _\ell }({{\mathcal {V}}}) - b_\ell \phi _{\mu _\ell }(-{{\mathcal {V}}})\Big ) = 0, \\&a_0-b_0+\sum _{\ell =1}^{K-1} \Big (a_\ell \phi _{\lambda _\ell }(-{{\mathcal {V}}}) - b_\ell \phi _{\mu _\ell }({{\mathcal {V}}})\Big ) = 0. \end{aligned}$$

  We now proceed as in the proof of Proposition B.1 by introducing the polynomial

  $$\begin{aligned} v\mapsto Q(v)&:= (a_0-b_0)\prod _{i=1}^{K-1}(1-\lambda _i v) \prod _{j=1}^{K-1}(1+\mu _j v) \\&\quad +\, \sum _{\ell =1}^{K-1} a_\ell \prod _{i=1,i\ne \ell }^{K-1}(1-\lambda _i v) \prod _{j=1}^{K-1}(1+\mu _j v) \\&\quad - \sum _{\ell =1}^{K-1} b_\ell \prod _{i=1}^{K-1}(1-\lambda _i v) \prod _{j=1,j\ne \ell }^{K-1}(1+\mu _j v). \end{aligned}$$

  This is a polynomial of degree $2(K-1)$ which admits the 2K roots, $\pm v_1, \ldots , \pm v_K$ (from above equalities). So it is the null polynomial. Picking the values $v=1/\lambda _\ell$ and $v=-\,1/\mu _\ell$, $\ell =1,\ldots ,K-1$, we deduce that $a_0=b_0$, $a_\ell =0$, and $b_\ell =0$, for $\ell =1,\ldots ,K-1$. Therefore, $Y_1=Y_2=a_0 {\mathbf {1}}_{{\mathbb {R}}^K}$.

• Consider an element in the range of ${\mathcal {H}}$, $Z={\mathcal {H}} Y$, with $Z=(Z_1\ Z_2)^\top$, $Y=(Y_1\ Y_2)^\top$, $Z_i\in {\mathbb {R}}^K$, $Y_i\in {\mathbb {R}}^K$, $i=1,2$. Then,

  $$\begin{aligned} \sum _{k=1}^K \omega _k({Z_1}_k + {Z_2}_k) = \sum _{k=1}^K \omega _k v_k \sum _{\ell =1}^{K} \big ((\zeta _1 \gamma _1)_{k\ell } {Y_1}_\ell + (\zeta _2 \gamma _2)_{k\ell } {Y_2}_\ell \big ). \end{aligned}$$

  Applying our assumption, we get

  $$\begin{aligned} \forall \ell , \qquad \sum _{k=1}^K \omega _k v_k (\zeta _1 \gamma _1)_{k\ell }=0, \quad \sum _{k=1}^K \omega _k v_k (\zeta _2 \gamma _2)_{k\ell }=0, \end{aligned}$$

  so, for any $Z=(Z_1\ Z_2)^\top \in \text{ Im }({\mathcal {H}})$, we have $\sum _{k=1}^K \omega _k ({Z_1}_k + {Z_2}_k) = 0$. The dimension of $\text{ Ker }({\mathcal {H}})$ is 1, so, thanks to the rank-nullity Theorem, equalities are as claimed in Lemma B.1. $\square$

C. Properties of eigenelements of VFP

This appendix is devoted to the proof of an analogue of Lemma B.1 for the VFP case under assumptions on the set of discrete velocities. We first define the useful notations. Let $\psi _\ell ^0$, $\ell =0,\ldots ,K-1$, be defined as in (5.10). Let us assume that assumptions (5.11)–(5.13) on the velocity quadrature hold. Therefore, there exists $\beta \in {\mathbb {R}}^K$ and $\gamma \in {{\mathcal {M}}}_{K-1\times K}({\mathbb {R}})$ such that

$$\begin{aligned} \begin{pmatrix} \beta ^\top \\ \gamma \end{pmatrix} = \Big (\psi _0^0({{\mathcal {V}}}) \quad \psi _1^0({{\mathcal {V}}}) \quad \cdots \quad \psi _{K-1}^0({{\mathcal {V}}})\Big )^{-1}. \end{aligned}$$

We introduce $\zeta _\ell := \psi _\ell ^0({{\mathcal {V}}})-\psi _\ell ^0(-{{\mathcal {V}}})$, and $\zeta := \big (\zeta _1 \ \ldots \ \zeta _{K-1}\big ) \in {\mathcal {M}}_{K\times K-1}({\mathbb {R}})$. Then we denote the matrix

$$\begin{aligned} {\mathcal {H}} = \begin{pmatrix} {\mathbf {I}}_K &{} \zeta \gamma - {\mathbf {I}}_K \\ \zeta \gamma - {\mathbf {I}}_K &{} {\mathbf {I}}_K \end{pmatrix}. \end{aligned}$$

Lemma C.1

With the above notations, if we assume that (5.11)–(5.13) hold. Then,

• $\text{ Ker }({\mathcal {H}})=\text{ span }\left( \exp \left( -\frac{{{\mathcal {V}}}^2}{2\kappa }\right) \right) =\text{ span } \Big (\psi ^0_0({{\mathcal {V}}})\Big )$,

• $\text{ Im }({\mathcal {H}}) = \Big \{Z= (Z_1\ Z_2)^\top ,\ Z_i\in {\mathbb {R}}^{K} \text{ such } \text{ that } \sum _{k=1}^K \omega _k ({Z_1}_k+{Z_2}_{k}) = 0\Big \}$.

Proof

We proceed as in the proof of Lemma B.1.

• Let $Y=(Y_1\ Y_2)^\top \in \text{ Ker }({\mathcal {H}})$, then

  $$\begin{aligned} Y_1-Y_2 = \zeta \gamma Y_1 = -\zeta \gamma Y_2. \end{aligned}$$

  By assumption (5.11), the family $\{\psi _0^0({{\mathcal {V}}}),\, \psi _1^0({{\mathcal {V}}}), \ldots ,\, \psi _{K-1}^0({{\mathcal {V}}})\}$ is a basis of ${\mathbb {R}}^K$, then, we may write

  $$\begin{aligned} Y_1 = \sum _{\ell =0}^{K-1} a_\ell \psi ^0_{\ell }({{\mathcal {V}}}), \qquad Y_2 = \sum _{\ell =0}^{K-1} b_\ell \psi ^0_{\ell }({{\mathcal {V}}}). \end{aligned}$$

  Simple computations using the definition of $\zeta$ and $\gamma$ and recalling that $\psi _0^0({{\mathcal {V}}})=\exp \left( -\frac{{{\mathcal {V}}}^2}{2\kappa }\right)$, give

  $$\begin{aligned} \zeta \gamma Y_1 = \sum _{\ell =1}^{K-1} a_\ell (\psi _\ell ^0({{\mathcal {V}}})-\psi _{\ell }^0(-{{\mathcal {V}}})), \qquad \zeta \gamma Y_2 = \sum _{\ell =1}^{K-1} b_\ell (\psi _\ell ^0({{\mathcal {V}}})-\psi _{\ell }^0(-{{\mathcal {V}}})). \end{aligned}$$

  Thus the equalities $Y_1=Y_2-\zeta \gamma Y_2$ and $Y_2=Y_1-\zeta \gamma Y_1$ imply

  $$\begin{aligned}&(a_0-b_0)\exp \left( -\frac{{{\mathcal {V}}}^2}{2\kappa }\right) +\sum _{\ell =1}^{K-1} \Big (a_\ell \psi _\ell ^0({{\mathcal {V}}}) - b_\ell \psi _\ell ^0(-{{\mathcal {V}}})\Big ) = 0, \\&(a_0-b_0)\exp \left( -\frac{{{\mathcal {V}}}^2}{2\kappa }\right) +\sum _{\ell =1}^{K-1} \Big (a_\ell \psi _\ell ^0(-{{\mathcal {V}}}) - b_\ell \psi _\ell ^0({{\mathcal {V}}})\Big ) = 0.\end{aligned}$$

  From assumption (5.12), we deduce that $a_0=b_0$, $a_\ell =0$, $b_\ell =0$, for $\ell =1,\ldots ,K-1$. As a consequence $Y_1=Y_2=a_0 \psi _0^0({{\mathcal {V}}})$.

• Consider an element in the range of ${\mathcal {H}}$, $Z={\mathcal {H}} Y$, with $Z=(Z_1\ Z_2)^\top$, $Y=(Y_1\ Y_2)^\top$, $Z_i\in {\mathbb {R}}^K$, $Y_i\in {\mathbb {R}}^K$, $i=1,2$. Then,

  $$\begin{aligned} \sum _{k=1}^K \omega _k({Z_1}_k + {Z_2}_k) = \sum _{k=1}^K \omega _k v_k \sum _{\ell =1}^{K} (\zeta \gamma )_{k\ell } \big ( {Y_1}_\ell + {Y_2}_\ell \big ). \end{aligned}$$

  Applying our assumption, we get

  $$\begin{aligned} \forall \ell , \qquad \sum _{k=1}^K \omega _k v_k (\zeta \gamma )_{k\ell }=0, \end{aligned}$$

  so, for any $Z=(Z_1\ Z_2)^\top \in \text{ Im }({\mathcal {H}})$, we have $\sum _{k=1}^K \omega _k ({Z_1}_k + {Z_2}_k) = 0$. The dimension of $\text{ Ker }({\mathcal {H}})$ is 1, so, thanks to the rank-nullity Theorem, rank$({\mathcal {H}})=K-1$, which allows to conclude the proof. $\square$

D. Some properties of exponential polynomials

1.1 D.1 Elementary proof of the Pólya–Szegö estimate

Hereafter, following [42, page 10], we establish by induction a simple bound on the number of real roots of an exponential polynomial; for various extensions, see [58, 59]

$$\begin{aligned} \forall n \in {\mathbb {N}}, \qquad f_n(x)= \sum _{i=0}^{n-1} P_i(x) \exp (\mu _i\, x), \qquad \mu _i \in {\mathbb {R}},\quad \text{ deg }(P_i)=k_i. \end{aligned}$$

We intend to show that, for any $n \in {\mathbb {N}}$, $f_n$ admits at most$N_n-1$ roots, where

$$\begin{aligned} N_n=\left( \sum _{i=0}^{n-1} (1+k_i)\right) . \end{aligned}$$

(D.1)

We use an induction on n:

• for $n=1$, the exponential polynomial reads $f_1(x)=P_0(x) \exp (\mu _0\, x)$ so it admits at most $k_0=N_1-1$ roots.

• Assume the property (D.1) holds for $f_n$, so that it admits at most $N_n-1$ real roots. Let M be the number ot real roots of $f_{n+1}$, and define

  $$\begin{aligned} \forall x \in {\mathbb {R}}, \qquad f_{n+1}(x)\exp (-\mu _{n}\,x)&=\sum _{i=0}^n P_i(x)\exp ((\mu _i - \mu _{n})\,x)\\&=P_n(x)+\sum _{i=0}^{n-1} P_i(x)\exp ((\mu _i - \mu _{n})\,x). \end{aligned}$$

  By the classical Rolle’s theorem for smooth functions, its $(1+k_n)\mathrm{th}$ derivative

  $$\begin{aligned} \forall x \in {\mathbb {R}}, \qquad g_{n+1}(x)&= \frac{d^{(1+k_n)}}{dx^{(1+k_n)}}[f_{n+1}(x)\exp (-\mu _{n}\,x)] \\&=\sum _{i=0}^{n-1} \frac{d^{(1+k_n)}}{dx^{(1+k_n)}}[P_i(x)\exp ((\mu _i - \mu _{n})\,x)], \end{aligned}$$

  admits at least $M-(1+k_n)$ roots. But since $g_{n+1}$ is an exponential polynomial to which (D.1) applies, it comes that

  $$\begin{aligned} M-(1+k_n) \le N_n -1, \qquad \text{ so } \text{ that } \quad M \le N_n + (1+k_n) -1 := N_{n+1} -1. \end{aligned}$$

1.2 D.2. Haar property for exponential monomials

Although the former estimate suggests that exponential polynomials do not constitute a Chebyshev T-system, non-negative exponential monomials do satisfy the Haar property on $(0,+\infty )$:

Theorem D.1

(Krattenthaller [43]) Let $(x_0,x_1,\ldots ,x_{n-1}) \in {\mathbb {R}}_+^n$, $(y_0,y_1,\ldots ,y_{n-1}) \in {\mathbb {R}}^n_+$ be non-negative with $y_0<y_1<\dots <y_{n-1}$. Moreover, let $(z_0,z_1,\ldots ,z_{n-1}) \in {\mathbb {N}}^n$ be non-negative integers with $z_0<z_1<\dots <z_{n-1}$. The generalized Vandermonde determinant,

$$\begin{aligned} \det _{0\le i,j<n}\left( e^{y_jx_i}x_i^{z_j}\right) \end{aligned}$$

(D.2)

vanishes if and only if two of the $x_i$’s are equal to each other.

The proof of this result relies on an expansion of the exponential and the use of Schur functions [44, 46, 47].