Discrete Pseudo-differential Operators and Applications to Numerical Schemes

Abstract

We consider a class of discrete operators introduced by O. Chodosh, acting on infinite sequences and mimicking standard properties of pseudo-differential operators. By using a new approach, we extend this class to finite or periodic sequences, allowing a general representation of discrete pseudo-differential operators obtained by finite differences approximations and easily transferred to time discretizations. In particular we can define the notion of order and regularity, and we recover the fundamental property, well known in pseudo-differential calculus, that the commutator of two discrete operators gains one order of regularity. As examples of practical applications, we revisit standard error estimates for the convergence of splitting methods, obtaining in some Hamiltonian cases no loss of derivative in the error estimates, in particular for discretizations of general waves and/or water-waves equations. Moreover, we give an example of preconditioner constructions inspired by normal form analysis to deal with the similar question for more general cases.

A Young Inequality for Convolution

Let $x = (x_n)_{n \in \mathbb {Z}^d}$ and $y = (y_n)_{n \in \mathbb {Z}^d}$ two sequences. We define $z = x * y = (z_n)_{n \in \mathbb {Z}^d}$ the sequence

$$\begin{aligned} z_n = \sum _{\begin{array}{c} p,q \in \mathbb {Z}^d \\ n = p + q \end{array}} x_p y_q, \quad n \in \mathbb {Z}^d. \end{aligned}$$

We also define for $p \ge 1$,

$$\begin{aligned} \Vert x\Vert _{\ell ^p} = \left( \sum _{n \in \mathbb {Z}^d} |x_n|^p \right) ^{\frac{1}{p}}. \end{aligned}$$

And we recall the following Hölder inequality: for two sequences x, y

$$\begin{aligned} \left| \sum _{k} x_k y_k \right| \le \Vert x\Vert _{\ell ^p} \Vert y\Vert _{\ell ^q} \quad \text{ for } \quad 1 = \frac{1}{p} + \frac{1}{q}, \end{aligned}$$

which is itself a consequence of the Young inequality for product: $\forall a,b\ge 0$ we have $ab \le \frac{a^p}{p} + \frac{b^q}{q}$. This Hölder inequality is easily generalized by induction to

$$\begin{aligned} \left| \sum _{k} \prod _{i = 1}^N x_k^{(i)} \right| \le \prod _{i = 1}^N \Vert x^{(i)}\Vert _{\ell ^{p_i}} \quad \text{ for } \quad \sum _{i = 1}^N \frac{1}{p_i} = 1, \end{aligned}$$

for any sequences $x^{(i)}$, $i = 1, \ldots , N$ with $N \in \mathbb {N}$.

Lemma A.1

For two sequences x and y indexed by $\mathbb {Z}^d$, we have

$$\begin{aligned} \Vert x*y\Vert _{\ell ^r} \le \Vert x\Vert _{\ell ^p} \Vert y\Vert _{\ell ^q}, \quad \text{ for } \quad 1 + \frac{1}{r} = \frac{1}{p} + \frac{1}{q}. \end{aligned}$$

Proof

Let us denote $z = x * y$, we have

$$\begin{aligned} |z_n|&\le \sum _{k \in \mathbb {Z}^d} |x_{n-k}| |y_k| \\&= \sum _{k \in \mathbb {Z}^d} (|x_{n-k}|^p |y_k|^q)^{\frac{1}{r}} | x_{n-k}|^{\frac{r- p}{r}} |y_k|^{\frac{r-q}{r}} \\&\le \left( \sum _{k} |x_{n-k}|^p |y_k|^q \right) ^{\frac{1}{r}} \left( \sum _k | x_{n-k}|^p \right) ^{\frac{r -p}{rp}} \left( \sum _k |y_k|^q \right) ^{\frac{r - q}{rq}}\\&= \left( \sum _{k} |x_{n-k}|^p |y_k|^q \right) ^{\frac{1}{r}} \Vert x\Vert _{\ell ^p}^{\frac{r-p}{r}} \Vert y\Vert _{\ell ^q}^{\frac{r-q}{r}}, \end{aligned}$$

where we used the generalized trilinear Hölder inequality for the decomposition

$$\begin{aligned} \frac{1}{r} + \frac{r -p}{rp} + \frac{r - q}{rq} = \frac{1}{p} + \frac{1}{q} - \frac{1}{r} = 1. \end{aligned}$$

Then we obtain

$$\begin{aligned} \sum _{n} |z_n|^r&\le \Vert x\Vert _{\ell ^p}^{r-p} \Vert y\Vert _{\ell ^q}^{r-q}\left( \sum _{k,n} |x_{n-k}|^p |y_k|^q \right) = \Vert x\Vert _{\ell ^p}^{r} \Vert y\Vert _{\ell ^q}^{r}. \end{aligned}$$

$\square$